\ . ‘. 5“ 45”.“. 4. y't.‘fi .M‘ This with-appli. versal stn and special Let , be a' class W1“ denote °f Power < We not K can be c \ (fi;/ 17/ ABSTRACT HOMOGENEOUS UNIVERSAL GROUPS BY Kenneth Keller Hickin This thesis contains a recasting and generalization- with-application of Jonsson's Theorem on homogeneous uni- versal structures, distinct from the notions of saturated and special structures. Let n be an infinite regular cardinal and let m be a class of algebras or relational systems of type (It. m< of power (:c. n denotes the class of m algebras generated by subsets We note that homogeneous universal structures of power n can be characterized by an "injectivity" property which we generalize to g:injectivity of complete chains of 7m algebras of arbitrary order types. In the case of a chain 1 < A.< B with two jumps, the definition is as follows: {1,A,B} < 7n is g-iniective for 721 iff, for all B,X,Y 6 W5“ such that X'< Y, B < Y and erE e‘m<” and for all embed- dings f: B 4 B such that f(B)r1A = f(8rWX), f extends to an embedding f:Y 4 B such that f(Y)r1A = f(x). This definition generalizes easily to arbitrary complete chains. BO M = < gen 0' f m subama algflmo) for any 0 of any co S.p. plus Theorem. need the 1 be chosen amalgams c p algebra HP to m Class: these Prep free QIOUp groups; fu A hat by Conside jumpftypes Kenneth Keller Hickin To construct n-injective chains, an assumption called B C the subamalgam property (s.p.l is used. 0’: ‘\\// is . A an m: amalgam iff A,B,C 6 m, d’= BLJC, and BfWC = A. B0 C0 ab = \\// is a subamalgam of 0' iff ab g.a' and A O BonA = ConA = A0. W has the s.p. iff for every 7R amal- gam a there exists M = algM(a) e 711 such that, for all 7” subamalgams do of a, we have algM(aO) fid= do and algM(ab) €‘m. To construct well-ordered n-injective chains for any ordinal <’(+ and to construct w-injective chains of any countable jump-type, we require that m w . . . 8 §1.3 The Construction of Hemogeneous Universal Algebras . . . . . . . . . . . . . . . . . . 17 §1.4 Some Groups of this Study: Free Algebraic Closures and Free x-Classes . . . . . . . . 27 CHAPTER II: DISCUSSION OF u-INJECTIVE CHAINS §2.l Definitions, Existence, and Uniqueness of n-Injective Chains . . . . . . . . . . . . . 49 §2.2 An Application of w-Injective Chains of Groups . . . . . . . . . . . . . . . . . . . 60 CHAPTER III: EXISTENCE AND SPECIAL PROPERTIES OF K-INJECTIVE CHAINS §3.l Proofs of the Existence and Isomorphism Theorems . . . . . . . . . . . . . . . 77 §3.2 w-Injective Chains of ULF Groups . . . . . 93 53.3 w-Injective Chains of Algebraically Closed Groups . . . . . . . . . . . . . . . . . . . 98 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 102 iv PREFACE OF NOTATIONS AND CONVENTIONS §O.l Set Theory Inclusion of sets is denoted by .g and proper inclu- sion by < . Qplpg 'gggdinals and Ordinals. A cardinal number n is the smallest ordinal number of cardinality n. An ordinal equals the set of its predecessors. x, A, 0 always denote infinite cardinals. a, B, Y always denote ordinals in statements such as 'a < n'. th The a cardinal is The smallest infinite car- w 0 a dinal will always be written w instead of mo. The cardinality (power, order) of a set S is |S|. I O + I The cardinal successog of n is n ; the ordinal suc- cessor of a is d-+l. The Generalized Continuum.§ypothesis (G.C.H.) asserts that for all u, n+ = 2”, the cardinal of the power set of n. n is regular if n is not the union of fewer than n sets, each of power (it. Thus, w is regular, and n+ is regular for all u. An ordinal Y is even if y = a-+2n where a is a limit ordinal or 0 and O g,n < w. All other ordinals are odd. 0.1.1 Functions. The identity function on a set S is ls. The notation 'f e g on 8' means that, for all s E S, f(s) = 9(3). we write f a l on 8 instead of f 2 1S on S. ‘Qplpg Chains. A.ghgig.is a set of sets which are totally ordered by set inclusion. If the members of a chain c. are indexed 21 ordinals, as in the statement c.= {Cale < n}, it is always assumed that, for all a < n, Ca S-ca+1' c. is continuous if, for all limit ordinals Y < n, C = UC . Y a = algM(S) is the smallest subalgebra of M containing S, that is, the sub- algebra of M generated by S. me ated 21.3 subset 2; power ()1. Thus m is called the trivial algebra of"m. If Q = o, then m has a trivial algebra ¢. Note that every isomorphism of m algebras extends IQ. 90.3 Group Theory. Let G be a group. If x E G, then the order of x is |x|. ix A‘g G will mean that A is a subgroup of C unless the context demands a different interpretation. If S is a subset of G, then ng(S) = w is regular, then x: has exactly one member, H”, up to 2. Every group of power x is embeddable in H“. (ii) The class HU$(£.f. groups) has exactly one member, Hw’ up to a. Every countable i.f. group is embeddable in H . w The group Hm was studied by Philip Hall [2] a little before Jonsson's second paper. A group in the class HUw(£.f. groups) is usually called a gpiyggggl_lggglly_£ip;§g ggggp, and we write ULF for this class. Hall gave a con- crete construction of Hw' He obtained it as the union of a chain of finite symmetric groups ...Rn < Rn+1“' where Rn+1 is the symmetric group on Rh, and Rh is contained in Rn+1 as its Cayley (right regular) representation. There is no countable group which is w-universal for the class of all groups because every countable group has at most w finitely generated subgroups, but there are 2w non-isomorphic 2-generated groups [14]. The members of HUQQM) can be defined by a single prop- erty which we will use in 51.3 to simplify the account of Jensson's theorem, and which we will generalize in 92.1. 1.1,? Qefinition of a x-Injective Algebra. An m algebra M is x-injective for m if, for all A,s e m“ with A < 3, every embedding f:.A 4.M extends to an embedding of B into M. The class of n-injective algebras for m is denoted INJK (7/1) and INJEIUII) = INJK(7)z)n771)‘ for 1 2 x. We define fix = INJKLJ) where J' is the class of all groups. 1.1.8 Proposition. For any class m, HUfi(m) g INJx(m). Proof. Suppose M e HUi(m), A < B are ‘m<” algebras, and f: A 4 M is an embedding. Since M is n-universal for m there is an embedding g: B 4.M. Since M. is n-homogeneous l for 'm, the isomorphism fg- : g(A) 4 f(A) extends to some q: 6 Aut M. Now, cpg: B -o M extends f. Hence M 6 INJK (7/1). Wfith.some weak assumptions on. m to be discussed in §l.3, we have INJ:(m) = HUtCM), but generally one cannot prove INJX (771) g HUn (771) because the required automorphisms of algebras of power >:¢ cannot be constructed. However, for groups we have 1.1.9 Proposition. (i) For all n > w, 3“ = x], and (ii) INJQ(L.f. groups) = ULF. £3993. Let G 6 9n (resp., INJw(1..f.)). To show that GEM“ (resp., ULF), suppose A and B are subgroups of G of power (It (s 2 w), and m 6 ISO(A,B). We must extend o to an automorphism of G. Let J = ng(A,B) and note that [J] < x. J is contained in a group H = (JIgt> such that for all a E A, t-lat = ¢(a) and such that |H| < K. To prove this, we can use Philip Hall's construction where t is a certain element of the symmetric group on J [14; p. 537], or, for n > w, we can use the HNN extension to be discussed in §1.2. Since G is n-injective, there is an embedding f: H 4 G such that f e 1 on J. Thus, for all a E A, 1 f(t)- af(t) = ¢(a), that is, m extends to an inner auto- morphism of G. A group G 6 Mg can be thought of as an "algebraic universe for group theory for groups of power <:¢". Jonsson's theorem says that there is exactly one such uni- verse up to 2 of power x (assuming the G.C.H and that x is regular). Such groups have intrinsic interest because their structure reflects, in many senses, the structure of all group theory. Any general construction possible in group theory can be applied within the groups H” and correlated with algebraic properties of these specific groups. The simplest example of this phenomenon is the previous proof where the existence of a particular group theoretical con— struction implies that all the automorphisms involved in the homogeneity condition for any G 6 Mi can be chosen inner. Because group theory is rich in general constructions, the homogeneous universal groups are an archetypal case of this phenomenon. ‘We will attempt to give further evidence of this in §l.2 by proving two non—obvious structure theorems for the groups H“, n > w. The purpose of this study is the presentation of a gen- eral algebraic construction which gives considerable infor- mation about the structure of certain homogeneous universal algebras. This construction involves generalizing the concept of a n-injective algebra, which we show in §1.3 to be a natural concept in JOnsson's original proof, to the concept of a u-injective chain 2; algebras (for the definition see 2.1.9). In Chapters 2 and 3 existence and uniqueness the- orems are proved for n-injective chains in a general alge- braic setting divorced from group theory, and some special properties of x-injective chains of groups are established. These special properties are developed to give an applica- tion of n-injective chains to a question of much interest in model theory - the so-called "spectrum problem" of con- structing many non-isomorphic models of theories. m we will use w-injective chains to construct 2 1 non-~ isomorphic groups belonging to the class HU:1(m) where m can be various classes of groups including the class of locally finite groups, classes obtained from algebraically closed groups (see 1.4.0 and 2.2.0), and classes admitting free amalgamations which we call "free w-classes" (see 1.4.20). This is not intended to be a definitive application of n-injective chains, but an illustration of their potential usefulness. These results are not new: it is the method of obtaining them which is. The existence of 2K non-isomorphic ULF groups of power u for all n > w was obtained by Macintyre and Shelah [12] using some deep techniques of model theory, and other ways to construct ULF groups are known [3] and [22]. Our application is not a complete redundancy since these groups have a special lattice property due to their construction as transfinite w-injective chains (2.2.1). Jill All}. Chapter 2 contains our results and many of the proofs. The remaining proofs are given in Chapter 3. In §l.4 we give some concrete examples of groups to which our applica- tion applies. we feel that the concept of u-injective chains is a direct extension of Jonsson's original idea, and that n-injective chains may be of general use in model theory. It should be noted that there is a well known general- ization of homogeneous universal systems. This is the con- cept of saturated models and chains of models due to Morley, vaught, and Keisler (see [1: Chapt. 11]). The concept of a u-injective chain is, of course, distinct from this and offers new possibilities.1 1I have asked S. Shelah if he ever saw this construction. He said that the only thing vaguely resembling it is a construction he used in an unpublished proof to construct recursive automorphisms of Boolean algebras, and he ex- pressed hope that n-injective chains would prove helpful in certain constructions. 91.2 Two Structure Theorems for H“, n > w, Obtained From the Free Amalgamation and HNN Constructions. we will first state the essential properties of these constructions. H K . 1.2.0 Definition. Let a'= ‘\V/ be a group amalgam. F The free amalgamated product of a, denoted 9P*(dl, is the group with presentation (a3 all relations of H and all relations of K). A sequence c1,...,c in a' is called n reduced if, for all i, ci {'F and successive ci, ci+l come from distinct factors of a; If a' is contained in a group G, and if la induces an isomorphism from gp*(a) to ngm), then we will say that a generates gp*(a) ip G. lpggl_ Qefinition. Let A and B be isomorphic subgroups of a group G and suppose o e ISO(A,B). Th2 HNN extension em = _o._f G relative 1:2 cp is the group with presen- tation (G,t; all relations of G and t-lat = ¢(a) for all a E A). t is called the stable letter. A sequence go, 6 e t1,gl,...,t n,gn (where 61‘11: 9166 and n21) is called reduced if there is no consecutive subsequence of the 1 1 form t" ,gi,t with gi 6A or t,gi,t" with 91 e B. If G and t are contained in a group H, and if lGuft} induces an isomorphism between Gcp and ng(G,t), then we will say that G and t generate the HNN extension Gcp i H. 1.2.2 Normal Form Theorem for Free Amalgamated Products [19: Th. III]. Suppose a' is a group amalgam, A is a group, and as A. Then a generates gp*(a) in A iff the product of every reduced sequence in a’ is non-trivial. 1.2.3 Britton's Lemma (Normal Form Theorem for HNN Exten- sions). [19: p. 614]. Let ch = be the HNN exten- e e sion of G relative to m. If go,t l,...,t n,gn is a e 6 reduced sequence and n z_l, then got l--- t ngn # l in gp(G,t). we will need the following two lemmas which are easy consequences of the normal form theorems. H K 1.2.4 If a'= \v/ is a group amalgam, G = gp*(a), and F U and V are subgroups of K such that vrwng(B,U) = l, then VngpG(H,U) = 1. 1.2.5 If G = is the HNN extension of G rela- cp tive to o E ISO(A,B), and if U is a subgroup of G such that UrWA = Ur1B = 1, then U and <”t> generate their free product in gp(G,t) and Gn = U. Our first structure theorem is ‘lpggp Normal Basis Theorem for H”. (G.C.H.). Suppose a > w is regular. Let F be the free group on [x,y}. For each ordinal Y < x, let wY(x,y) E F be a reduced word of length at least 2 such that (me> is a maximal cyclic subgroup of F. 10 H” has a generating set [aY,bY[Y < x} such that for ' F = b is free on a b and all Y < K: (3;) Y , (3'._i_) if 1r is a product in which non-trivial elements of HY alternate with non-trivial elements of Fy’ than n = 1 in an only if n contains a consecutive subword of the form wy(a ,bY)3, j #'0. Y 1.2.7 Corollary. (3) If 0 is a non-trivial reduced word in the free group with free basis {aY’bYlY < n], then o==1 in H“ only if a has a consecutive subword of the form j . . wy(ay,bY) , j #'0, and for some 5 < Y either aB or bB occurs in o. ()3) Let A= and B=. Then A and B are free subgroups of H” = , and if i j 11 no word wY has the form (x y ) , then for all 1 #’u e A and 1 #‘v E B, (11> and (‘7) generate their free product in H . n Discussion. The theorem has been stated in the manner most convenient for proof. Part (a) of the Corollary is actually a sharper formulation of the theorem describing the “freeness” with which the generators [aY,bY|y < u} generate H”. Part (b) is the most quotable result, saying "Ha is generated by two disjoint free subgroups which interact pair- wise freely“. Proof of (a). Suppose o = 1 in H“. Let y be the largest ordinal for which aY or bY occurs in 0. Then 0 = hof1 fnhn’ n 2 1, where 1 7! fi 6 (ay,by> = FY , hi is a reduced word in the free group G on {aB’bBIB < y] and 11 h .,h are non-trivial in G. Since 0 = l in H”, 1,00 part (ii) of the Theorem (with the above decomposition of n-l 0 used as w) implies that either some hi which is non- trivial in G is trivial in H“ or some fi = w$.. ‘We can rule out the former case by induction, and assume fi = wg. Since fi # l in H” by part (i), some hj # 1 in G and hence some aB or b5” 6 < y, occurs in hj and hence in a. Proof of (b). That A and B are free groups follows from part (a) and the assumption that each word wY has length at least 2. If 1 # u E A and l # v e B and w is a product where powers of u alternate with powers of v, then the only consecutive subwords of w belonging to <(a ,by>’ for any Y can be of the form (aibgril unless Y ue and v€° So will in H” by part (a). gpoof of the Normal Basis Theorem for H”. Let [zy|0<:Y<(n, Y even] be a list of all the elements of H“. we will con- struct the sets {ay,by] inductively in such a way that (i) and (ii) of the Theorem hold and so that, if y is even, then 2H e HY = (aB’bBIB for all even H < y. This guarantees that [ay,by[y < n} does generate H“. Assume {aB,bB|B < y] have been chosen and y is even. Let p be minimal such that 2“ z'Hy. Note that 'Hy' < n and that ”,2 y by the inductive assumption. Let J = <;u;v> be a group such that 2H E J, u and V’ have infi- nite order, a= J) is an amalgam, and < 2 H) 12 1.2.8 n = 1 = (u) n <2“). (We can define J = (V) *< z“) where v is a formal letter, and let u = v-lzuv). Using n—injectivity of H“, let f be an embedding of 9P* (0) into H” such that f I 1 on ° Intuitively, f reproduces the amalgam a in H”, with (szH) fixed, so that f(a) generates gp*(f(a)) in H“. To simplify notation we will assume now that d n = l by 1.2.4 and 1.2.8. Now choose elements a b H so that F = a b is Y, Y e K y < Y. Y> F (d) freeon [a b} u=w(ab)anda=Y isan Y’Y’ YY’Y’ l <\/> u amalgam which generates gp*(d1) in H”. This can be done using n—injectivity as before. Since < u) n (Hy) = 1, the Normal Form Theorem 1.2.2 implies that FY has the property (ii) of our Normal Basis Theorem since u = wY(ay,bY). the that 1.2.4 and 1.2.9 imply 1.2.10 fl = 1. Again using x-injectiVity, choose aY+l’bY+1 E H“ such that FY+1 = is free on {aY+l,bY+1}, v = wY+1(aY+1, FY+1 (41> and a2 = \/ is an amalgam which generates < V > 9P*(d§) in H”. As above, 1.2.2 and 1.2.10 imply that (ii) by+1) ’ holds for HY+1 = < = = HY+2’ and this completes the ,Fy> and FY+1° We also have zY e inductive step. 13 Before stating our second theorem we need a definition. 1.2.11 Definition. Let (T,\) be a tree and for each or- dinal a, let Ta be the set of vertices of T at level a. If a is an infinite cardinal, we will call T a uétree provided (i) For all a < a, [T < K, a l (ii) For all a < K, x 6 Ta’ and B, a < B g’x, there exists y 6 TB such that x < y, and (iii) For all u # v 6 Tn’ there exists a < n and x 6 Ta such that x < u and x 4 v. Thus the members of Tn are in correspondence maximal chains in T(x = LflTala < x}, and each maximal chain in T has power a. we have not specified the size of Tx’ w is regular and (T,<) is a n-tree. For every u e Tn’ let T(u) = [x E T (in particular [M(u)r1M(v)l g n), and 14 (iii) Let u 6 Tu and a,b E HK with |a| = [bl = w, and \aa> n M(u) = \13> n M(u) = 1. Then there exists x 6 M(u) such that x-lax = b. The third property gives the key to the construction which uses HNN extensions. ggpgf. Let [zp|0 g,“ < n, u an even ordinal} be a list of Hu' Let q 6 Ha be a fixed element of infinite order. we will construct f inductively on each level TY of T, y g n. The construction will have the following prop— erties. If v e Ty, Y.S u, let T(v) = [u E T|u < v]. Assume f is defined on T, (3) For all u E Ty, <<1> n M(u) = l, (g) Suppose y is even, g < y is even, v 6 TY, v > u e T“, and ZH {M(u). Then q E ° (e) Suppose y is even, p < y is even, u E T , u u < v e TH+1’ [zu] = w: and <2“) n M(u) = 1. Then q = f(v)"lzuf(v). Suppose these properties hold for all y < n. ‘We will check that (i)-(iii) of the Theorem hold. Condition (iii) is an easy consequence of (e) since (e) guarantees that a and b are both conjugated to q by members of f(T(u)). Condition (ii) is immediate from (b). Since the freeness of each M(u) is asserted in (a), we need only prove that each M(u) is maximal in H“. For this we need a small lemma. 15 1.2.13 For all u 6 Tu and y e H“, there exist s,t 6 Ha such that |s| = |t| = w, (s) nM(u) = (t) n M(u) = l, and y e . lggggfi. By x—injectivity, let a,b E HK such that [a] = ‘b‘ = w and G) ® exists in H“. We claim that either the pair of subgroups {,} or the pair [,} has the property that both of its mem- bers intersect M(u) trivially. For otherwise M(u) would contain a free abelian subgroup of rank 2 contrary to the freeness of M(u). Now take 5 and t to be the generators of this good pair above. Now to show M(u) is maximal in H“, suppose x e Hn- M(u). We will show HK = . Since x = 2H for some (1, applying (d) for Y > p, we have q 6 < < . Hence M(u) is maximal, and (i)-(iii) follow from (a)-(e). The Construction of f pp To and T even. p+l’ P Assume f is defined on T

and note P that [J] < n. By n-injectivity, choose f(s) e H“ so that ]f(s)| = w and J and. <;f(s)> generate their free product in H“. Properties (a)-(c) are easily checked in the new cases where u or v is a successor of t in TP+1 by appeal to normal form in. J *<;f(s)> and the inductive as- sumptions. Thus f can be defined on T . we now assume f is defined on Tspp and on a subset U of Tp+l and that (a) - (e) hold for y = p +2 and for rel- evant u,v e Tsy. We must define f on some t e Tp+1-U. Let G = < f(TSp UU),zp,q>. Case 1. Izp‘ =w 21g nM(s) =1 mpg t>seTp. In this case, we must take care to satisfy (e) for Y = p-tz, p = p, u = s, and v = t. Note that (d) follows easily from this where y = p-tz, p = p, and v is a successor of t in Ty’ Now n = n M(t) = 1 by the previous construction, using normal form in Je< f(s) >. Let to E ISO(< zp>a) be such that cp(zp) = q. By a- injectivity, choose f(t) 6 Hn so that G and f(t) gen- erate the HNN extension GT with stable letter f(t) in H“. Thus (e) is satisfied, and (a)-(c) in the new cases where u or v is a successor of t in TY are all easy consequences of the corollary 1.2.5 to Britton's Lemma and our inductive assumptions. Case 2. For some i > 1, z: E M(s), but zp £'M(s). In this case (e) is vacuous, but we must take care that (d) 17 holds where p = p, y = p-t2, and v is a successor of t in Ty' Let y = zpf(s). By the previous construction of f(s), we have |y| = w and (y) n = (y) n M(t) = l by normal form in J*< f(s)) . Let cp E ISO(,) be such that cp(y) = q. By n-injectivity choose f(t) 6 HK so that G and f(t) generate the HNN extension Go with stable letter f(t) in H”. New (a)- (c) hold as in the previous case, and (d) holds since q 6 < < w if s < v 6 T9”. Case III. 2p 6 M(s). This is the trivial case. We need only satisfy (a)-(c), and this can be done by choosing f(t) so that G and < f(t) > generate their free product in H“. This completes the proof of the Maximal Subgroup-Tree Theorem. §1.3 The Construction of Homogeneous Universal Algebras. Our sketch of Jonsson's construction will be similar to [1: Chapt. 10] but we will use the idea of x-injectivity to unify the presentation. The construction can be motivated by asking several questions about any classof algebras m. (1) What minimal property must ‘m have if there exists a n-injective algebra for m? (2) What properties must ‘m have in order that any two mem- bers of INJ:(m) be isomorphic? 18 (3) What properties will guarantee that INJ:(W) = HU:(m)? Are any further properties needed to actually construct a n-injective algebra for AM? In answer to question (1), we want a relative version of u-injectivity which does not make reference to any parti— cular algebra. 1.3.0 The n-injective property for m. ‘m provided given any three MS” A < A and given an embedding with B S_B and an embedding tends f. (Note that any M 6 This is defined as follows. is x-injeCtive algebras A, A, and B ‘with f: A 4 B, there exists B E‘m f: A 4 B such that f ex- INJK(7I() will serve as B provided B g_M). ‘m is injective if ‘m is n-injective for all n. The injective property is a concrete version of what is usually called the "amalgamation property” and defined as follows. 1.3-1 The amalgamation property for m [1: p. 203]. If A,BO,Bl e‘m, f0 13 an embedding of A into 30’ and f1 is an embedding of A into B1, then there exists C E‘m and embeddings go of B0 into C and 91 of B1 into C such that the following diagram commutes: B o f/\ O . go 19 This is equivalent to the injective property by identi— fying A with a subalgebra of B = A and BJ. with a sub- 0 algebra of C = B. To answer question (2), the idea of constructing an isomorphism.between n-injective algebras fer ‘m is quite simple because n-injectivity can be used to enlarge any par- tial isomorphism between ‘m<” subalgebras. Specifically, suppose M,N e INJu (771), A,B 6 7719‘ are subalgebras of M,N respectively, f E ISO(A,B), and x 6 Ma-A. we can use x- injectivity of N to extend f to an embedding defined at x provided there exists an ‘m<” algebra C ’such that < A,x> g C s M. Alternatively, if y e N-B and there is an 76" algebra D such that < B,y> g D g N, we can use n—injectivity of M to define an embedding g of D into -1 on f(A), and then define f1 = 9'1. M such that g a f Then fl is an extension of f to C = g(D) and we have y e f1(C). In steps like these, both the domain and the range of the partial isomorphisms can be enlarged to include arbitrary elements. But to guarantee the existence of C and D we need the 'u-local property“. Lu £112 u-local property for 71. Suppose M 6 7n and x is a subset of M. with [K] < n. Then X‘ is contained in some ‘m<” subalgebra of M. Thus, the u-injective and n-local properties seem to allow us to build an isomorphism m of M onto N, provided [M] = [N[ = x, by defining m piecemeal in a back-and-forth 20 manner on ‘m<” subalgebras of M. In fact, two more prop- erties are needed to accomplish this. We must guarantee that, during the construction of w, the domain of each partial o is in m. For this we need 1.3.3 The n-inductive property. m is x-inductive if, for all chains 6' of m algebras with [C] 3.x, we have LC'E m. m is inductive if ‘m is n-inductive for all u. Finally, we need a property which guarantees that the isomorphism m can be started, that is, that M and N have isomorphic m subalgebras. 1.3.4 The comparison property for m. If A,B E m, then there exist A,B E m with A.g A and B g,B such that there exists C e m which is embeddable in both A and B. The role of this property appears in the following proof. 1.3.5 Lemma. If ‘m has the comparison and u-local prop- erties, then every u-injective algebra for m' is u-universal for ‘M. 135993. Suppose M E INJx(7I() and A 6 7/6". We must embed A into M. Using the n-local property, let B be any m6” subalgebra of M, and let A,B, and C be as in the compar- ison property and assume C S.A. By the n-local property we can also assume A,B,C e‘m<”. Let h be an embedding of C into B. Since M is n-injective there is an embedding f: B 4.M such that f a 1 on B, and there is an embedding 21 g: A 4.M which extends fh: C 4 M. gPA is the required embedding of A into M. The theorem concerning isomorphisms between x-injective algebras can be stated as follows. The previous comments hopefully make clear how the proof goes. This lemma is similar to [1: Isomorphism Theorem, p. 207]. 1.3.6 Isomorphism Lemma. Suppose m has the n-inductive and n-local properties. (3) If M,NEINJKWZ), A w and C has a subchain .9 such that C = up and [.D[ < x, because otherwise every S of (*) is contained in some A 6 C. Now suppose S < C with [S[ < a [is given. Let D e .9. We will construct Sl < C such that 8.3 S:l E 775" and 81 0D Ne W‘M. Put T = 0 S and assume To 3 - - . g Tn < C have been chosen and Tn e 775". By the u-local property of 7): there is some 1% g D 26 such that xne 776" and Tn nn g 1%. Again by the n-local <1 property, choose Tn+1 6 771 such that Tn an S Tn+1 g C. Put 8 = UT . Then [S [ < it since u is regular. 1 n l n20 Since 7/) is x-inductive we have Sl nD = an EMPM. Now n>_0 assume .3 = [Da]a is an ordinal (p] where p < x. We construct algebras S“, p < p, as follows. Let S0 = S, and construct Sa+l from so in the manner above, replac- ing S by Sq, D by D and S by S At limit a+1’ 1 CH1 ' a, put S“ = USB . The regularity of n insures that B m, but no such assumption is needed to apply Theorem 1.3.12. Another way to look at the content of Theorem 1.3.12 is that an algebra Me‘mx which is n- injective for ‘miM is determined up to a ‘by its m3” sub- algebras; that is, if M0 and M1 are any two such, and M0 5! M1, then, for i = 0 or 1, some 775" subalgebra of M1 is not embeddable in Ml-i' §1.4 Some Groups of this Study: Free Algebraic Closures and Free n-Classes. Let .3 be the class of all groups and u the class of all groups of order (n which are m-homogeneous for .3: that is, by Theorem 1.3.12, 6 e u iff G e magma) = Hugcgic) iff, given any o E ISO(A,B) where A and B are f.g. subgroups of G, o extends to an automorphism of G. For example, every divisible abelian group has this prop- erty. Theorem 1.3.12 says that any such group is determined uniquely in u by the set of its f.g. subgroups. An impor- tant subclass of u are those groups G such that every‘ m as above extends to an inner automorphism of G. we will call such a groupiipppp-homogeneous'gpp " such that y and 2 have infinite order and b = yz, and there is an extension‘ K: = 1 1 K* * such that c = (u- Equ- Ev) has infinite order. K1 can be defined by an easy free amalgamation. Let A = K1 *KKZ and let J = (A, t,s> be a group such that t'lct = y and s'lcs = z. , J can be defined by a sequence of two HNN extensions.. The existence of J implies that 1 1 -1 the set of words W1J{yzb-1,t- cty- ,s csz-l] with variables x1,...,x u,v,y,z,s,t is consistent over G, and this set n’ . forces r # 1 since the non-trivial element b e G is in the normal closure of r. we will now check our initial assertion. 1.4.3 Every a.c. group is inner-homogeneous for ng ([11: Lemma 1]). Proof. Suppose G is an a.c. group, A= and B are f.g. subgroups of G, and o 6 ISO(A,B). The set of relations x-laix = ¢(ai), l g_i gDn, is consistent over G 30 since they are solvable in an HNN extension of G. Hence 1 for some t e G, t- ait = ¢(ai), l g_i‘g,n. Thus a countable a.c. group is determined uniquely in the class of countable a.c. groups by its set of f.g. sub- groups. Algebraically closed groups have been and continue to be the subject of research since Scott introduced them [18], [11], [4], [22]. The main technique of this study (n- injective chains) is not directly applicable to arbitrary a.c. groups, but can be applied to a natural class of a.c. groups, one of which is discussed by Macintyre with the method of forcing [11: Theorem 8, p. 81]. We will now discuss some members of this class. lpgpg_ pefinition o§_§£ee extensions of groups. Suppose ‘W = W(A,x) is a set of words over a group G which involve only elements of the subgroup A g_G and the variables :x. The .2522 extension p§_ G. 21, W, denoted 6* = G*(W), is the group ‘with presentation [(G,x: all relations of G and W'= 1). ‘We say that G* is a finitary‘gppg extension (f.f.e.).p§, G if x. and W’ are both finite and W' is consistent over .0. The structure of G* is somewhat clarified by the next observation. 1.4.5 Lemma. Let .w'= W(A,X) be a consistent set of words over G and put E = G*(W). Then, G is a subgroup of E, 31 G A*(W) 9PE(A:X) = A*(W), G nA* (W) = A, and the amalgam a = V A generates gp*(a) in E. Proof. To show that G < E use the universal mapping prop- erty of presentations and the fact that W’ is consistent over G. The remaining assertions follow from the fact that gp*(a) has the same presentation as E. lpgpg Definition of the free extension prOperty. Let G < K be groups. K has the free extension property (f.e.p.) 232; G if, for every finite subset Sig K and for every finitary free extension J* of J = (6,8), there is an embedding. f of J* into K such that f a 1 on J. ‘We say that K is an f.e.p. group if K has the f.e.p. over 1. Notice that every f.e.p. group has the f.e.p. over each of its f.g. subgroups, and is therefore prima facie an a.c. group. 1.4.7 Lemma. If K has the f.e.p. over G, then K has the f.e.p. over every subgroup of G. .gppgg, Suppose K. has the f.e.p. over G and A.g,G. Let S be a finite subset of K, J = , and let J* (W) be a f.f.e. of J. By Lemma 1.4.5 we have J*(W) g *(W) = Q, and by 1.4.6 there is an embedding f of Q into K such that f s 1 on (6,8). Now theirestriction of f to J*(W) satisfies 1.4.6 proving that K has the f.e.p. over A. 32 1.4.8 Lemma. If B is a f.f.e. of A and C is a f.f.e. of B, then C is a f.f.e. of A. Proof. Let B = A*(Wl) and C = B*(W2) and let X1 and X2 be the sets of variables involved in WI and W2 re— rela— spectively. Then C has the presentation (A,X LJX 1 2’ tions of .A, wlLJWZ) showing that C is a f.f.e. of A. 1.4.9 Definition of a free algebraic closure of a group. Suppose G g K are groups. K is called a free algebraic closure (f.a.c.) of G prOvided (i) K has the f.e.p. over G, and (ii) For every finite subset S < K, there is a f.f.e. G* of G and an embedding f of G* into K such that f a 1 on G and s < f(G*). This concept has apparently never been studied. The universal algebraic closures of Jonsson [10] are much dif— ferent. 1.4.10 Theorem. (a) Every group G. has a f.a.c. K with [K] = max(|G],w). (p) If G < H < K are groups, is a f.a.c. of G, and H K is a f.a.c. of H, then K is a f.a.c. of G. (g) If G < H < K are groups, H is a f.f.e. of G, and K is a f.a.c. of H, then K is a f.a.c. of G. V (g) If G is a f.g. subgroup of an f.e.p. group F, then there is a f.a.c., K, of G with G < K's F. 33 (e) If G is a countable group and K1 and K2 are count- able f.a.c.'s of G, then there exists 3 E ISO(K1,K2) such that $ 2 1 on G. Proof of (a). This proof follows a suggestion of Professor Sonneborn. It is more natural than my original proof. Let H be any group and let T = %(H) = [W5]a E I} be a "complete" set of consistent finitary sets of words over H in the sense that if H*(W) is any f.f.e. of H, then there is some a E I such that the sets W and Wd are identical under some one-to—one correspondence of their vari— ables. Let Xfi be the set of variables involved in Wu and put XW) = UXa . We assume that these variable sets GEI are pairwise disjoint; that is, if a t B e I, then XerXB== ¢L We define the group H*(%) to have the presentation (H,Xa,a E I: relations of H,Wd,a e I), and we observe 1,...,Xn are the respective sets of variables, then the subgroup J = of 1.4.11 If Wl,...,Wh E W’ and X H*(W) has the presentation P = (H,Xl,...,Xh; relations of 13,Wi,...,Wh). In particular, J is a f.f.e. of H. The proof of 1.4.11 is that we can construct H*(W) in two steps: first presenting the group P as above, and then adding all the other letters Xd, a 6 I with Xa # Xi, 1 g_i g_n, with relations W . An application of Lemma 1.4.5, a using the disjointness of the sets X5, shows that P = .I < H*CV). 34 Now let G be any group. We will iterate the above construction m times. Specifically, let #1 = T(G) and define G1 = G*(Wl); having defined Gn’ let ”n+1 = T(Gn) = * . z - and put Gn+1 Gn(wn+l). Finally, put K lJChl. In this n41 construction we assume that the sets of variables X(%n), n 2_1, are pairwise disjoint. For any group H, simple cardinal arithmetic shows that IT(H)[ = max(]H[,w) = [H*(W)], and thus, in our construction, [K] = max([G],w). Now we must check that K is a f.a.c. of G. To show that K has the f.e.p. over G, let S be a finite subset of K, let J1=<:G,S>g and let J*(W) be any f.f.e. of J. Now J < Gn for some n 2,1, and there exists Wb E ”n+1 such that W and WE are identical under some one-to-one correspondence of their variables. Thus there exists f e ISO(J*(W),J*(Wd)) with f a 1 on J. Since J*(Wa) < G;(Wa) < G;(”n+1) by 1.4.5 and 1.4.11, f meets the condition of 1.4.6. To prove (ii) of 1.4.9, assume inductively that if T is any finite subset of Gn’ for some n 2_l, then T is contained in some f.f.e., G*(U) < Gn’ of G. Note that, if n = 1, this is immediate from 1.4.11. Let S < Gn+1 1x; finite. There exist Wl,...,Wh E T(Gn) with respective variable sets X1,...,Xfi such that S <_<;Gn,X1,...,Xm)n Now the words WlLJ...LJWh involve only finitely many ele— ments T of Gn’ and each Wi’ i.g i §.m, is a set of words over . By the induction hypothesis T < G*(U) < Gn’ 35 and, by 1.4.5 and 1.4.11, C = < is a f.f.e. of G*(U). By Lemma 1.4.8, it follows that C is a f.f.e. of G, and since 8 < C we have proved (ii) of 1.4.9. Hence K is a f.a.c. of G. [gpppf of (b). Suppose K is a f.a.c. of H and H is a f.a.c. of G. By Lemma 1.4.7, K has the f.e.p. over G. So we need to check (ii) of 1.4.9. Let S < K ‘be finite and choose H*(W) = H*, a f.f.e. of H, and an embedding f of 11* into K such that f s 1 on H and s < f(H*). Let x. be the set of variables in W’ and let T be the (finite) set of elements of H which occur in words of ‘W. Choose G*(U) = 6*, a f.f.e. of G, and an embedding g of G* into H such that g e 1 on G and T < g(G*). Let Y be the set of variables in U. New J = is seen to be a f.f.e. of G by applying 1.4.5 and 1.4.8 (via the isomorphisms g and f). Since 8 < J, we have now proved that K is a f.a.c. of G. Egoof of (c). The proof is similar to (b), with the simpli- fication that g(G*) = H is fixed. ggpof of (d). Let G be a f.g. subgroup of the f.e.p. group F. we W111 construct a chain ---Gn < Gn+1 <... of f.g. subgroups of F with G = so and the additional properties 1.4.12 For each' n12_0 there is a f.f.e. G; of Gm and * m E ISO(Gm,Gm+1) such that w s 1 on Gm, and 36 1.4.13 For every n 2.0 and for every W E T(Gn) (as in part (a)), there exists m > n such that G; = G;(W) where G; is the f.f.e. of 1.4.12. If Gm has been constructed for some m and is f.g., then, once we choose 6;, we can find Gm+l < F by a direct application of the f.e.p. of F. Since each of the sets T(Gn) is countable, there is no difficulty in choosing the G; so that 1.4.13 holds. Thus we can construct K = UGI1 . Q21 so that 1.4.12 and 1.4.13 hold. The proof that K is an f.a.c. of G is very similar to the proof in part (a), so we will omit it. In fact, this was our original proof of the existence of free algebraic closures. Proof of (e). First we need 1.4.14 Lemma. Suppose G < H < G* for some f.f.e. G* of G and H = with S finite. Then there is a f.f.e. H* of H and f e ISO(H*,G*) such that f s 1 on H. Proof. Suppose G* = (G,X; relations of G, W). For each s 6 S let u(s) e G* be an expression for s as a product of elements of G and X. Let U = [s-lu(s)[s E S}. Let K’ be letters in one-to-one correspondence with X and let W’ and U be the sets of words obtained by substituting the i’ for the X. Define H* = (H,i; relations of IL 'W,U),and for every 2 E H* define f(2) = z e 6*. It is easy to see that f e ISO(H*,G*) since every relation of H* is satis— fied in G* (and vice versa) under the correspondence i..x, 37 and f E 1 on H because the relations U guarantee this. This completes the proof of 1.4.14. New suppose K1 and K2 .are countable f.a.c.'s of G. We will build m E ISO(K1,K2), with) T a 1 on G, induc- tively on a chain of subgroups of K1 which are f.g. over G. The proof is similar to that outlined for the Isomorphism Lemma 1.3.6. we will give the essential step. Suppose H = < K1 with T finite and an embed— ding h of H into K2 has been defined such that h a l on G. (h is an approximation to m). We must be able to extend h to an embedding of A = < K1 into K2 where F is an arbitrary finite subset of K1' Using 1.4.9(ii) (with S = T(JF), let G* be a f.f.e. of G and 9 an em- bedding of G* into K such that g _ 1 on G and A < 1 g(G*). By Lemma 1.4.14, there is a f.f.e. H* of H and f E ISO(H*,g(G*)) with feel on H. Since K2 has the f.e.p. over G (letting J = h(H) in 1.4.6), there is an embedding e of H* into K2 such that e s h on H. New h = of.1 embeds g(G*) into K2 and extends h. Hence EPA is the desired extension of h. (See the diagram below). 5 9(G*)= f(H*)—>e(H*) ‘A<:;\~\;:\ h ///' H--—%h(H) \G/ At the next stage of the construction, we will reverse -1 the procedure and extend h in K2. Since K1 and K2 38 are countable, an isomorphism m with domain K1 and range K2 can be built in this way, and we have m a l on G since this holds at every stage. 1.4.15 Remarks on generalizing the concepts of free exten- sions and free algebraic closures. These concepts can be defined in any class m in which algebras have presentations, that is, in which there exist free algebras with the universal mapping property. This is the case if m is any variety of algebras. But, to prove Theorem 1.4.10 for a variety m, the following fact is needed to replace Lemma 1.4.5: if W is a consistent set of words over M e m and Mpg N 6 m, then W is consistent over N and M*(W) g N*(W). This can be easily proved if m is injective (refer to 1.3.0). Letting A = M, A = N, B = M*(W), and f = 1M, the proof would be to choose B and f as in 1.3.0. Put J = alg§(M*(W),f(N)). Then J is a homomorphic image of N*(W) by a homomorphism extending fLJlx where X is the set of variables of W. Thus N g N*(W) and M and X generate M*(W) in N*(W). So Theorem 1.4.10 holds in any injective variety. For example, the variety u of abelian groups is injective. The f.a.c. ip m of every ”Sm group is the divisible group whose torsion-free rank and p-rank, for every prime. p, equals w. This group is also the (unique) member of HU‘”(91) . (1) 1.4.16 Definition. A group K is a f.a.c. group if K is a f.a.c. of one of its subgroups. 39 We have introduced two natural classes of a.c. groups- the class of f.e.p. groups and the class of f.a.c. groups, which is contained in the former. Several observations can be made. 1.4.17 Theorem. (3) There are 2w non-isomorphic count- able f.a.c. groups; (2) Let Kb = the countable f.a.c. of the trivial group. K0 is embeddable in every f.e.p. group: every finitely presented group P is embeddable in ‘Kb, and, hence, Kb is the countable f.a.c. of P. The proof of (a) is a simple counting argument: there are 2w non-isomorphic f.g. groups, but a countable f.a.c. group has only w f.g. subgroups. That K0 is embeddable in every f.e.p. group is immediate using 1.4.10(d). If P is finitely presented, then P k a f.f.e. of the trivial group: so P is embeddable in every f.e.p. group, and a simple application of 1.4.14 shows that Kb is the countable f.a.c. of P. The countable a.c. group G, discussed by Macintyre [11; Theorem 8], such that its f.g. subgroups are exactly the f.g. recursively presented groups, is none other than Kb. Every f.g. subgroup H < K.0 is contained in a finitely presented subgroup (a f.f.e. of l), and hence H is recursively pre- sented. 0n the other hand, the celebrated theorem of Graham Higman asserts that every recursively presented group is a subgroup of a finitely presented group [5], and hence is em- beddable in KO’ 40 A countable f.e.p. group, K, exists which is not a f.a.c. of any of its f.g. subgroups. To construct K we need only arrange that, for every f.g. G < K, there is some f.g. B < K such that G is not embeddable in the countable f.a.c. of G. However, K might be a f.a.c. of a non-f.g. subgroup. This creates a prdblem we have not been able to solve. lg4.l8 Problem. Construct a countable f.e.p. group which is not a f.a.c. group. We now return to our main topic. If K is a countable f.e.p. group, then [J =,31K is an w-class by Theorem 1.3.12. In particular, J' has the m-injective (i.e., amalgamation) property. In fact, .J enjoys a much stronger amalgamation property which will serve to motivate the main concept of this section. 1,4.19 Theoppm, Suppose K is a f.e.p. group, ,J =.&[K, A B and a = \l/ is an J, and G = < f(A),g(B) >. If 1 g_i g_n, let wi = x-1f(ei)x[g(ei)]_1, and let h be an embedding of G* = G*(wi[1 g i g n) = into K such that h e l on G. Now G and h(x) = y generate in K the HNN extension of G relative to m 6 ISO(f(E),g(E)) 41 with ¢f(ei) = g(ei), 1 g_i g_n, because 6* is the HNN extension. It follows easily from Britton's Lemma (1.2.3) y‘1f w. If H is any set of primes, we have as examples (1) 3 = the class of all groups in which every element of finite order is a n-element, and (g) 3 = the class of all groups in which every periodic n-subgroup is locally finite. Both of these classes 3 contain gp*(a) whenever a' is an 3 amalgam. The proof of this fer (1) is quite easy 42 [19; Theorem II], while for (2) we can use the subgroup theorem for amalgamated free products [6; p. 228]. Every free u-class is a x-class. The comparison prop- erty follows from (i), while condition (iv) implies that every 3 m, then HUtCM) has a unique member up to a. In particular, every free n-class m has these properties. .gggpf. we need only show that m” #’¢. For this it suffices to show that, for every M.e m<“, there exists N'e m with M < N because we can then use the n-inductive property to obtain a member of"m”. we can assume that M has a proper m subalgebra E by the comparison property and assumption M M 1 (c). Now the amalgam \\/’ , 'where M1 e.M, is contained E in some N'e m by (b). 44 Jonsson remarked in [8] that assumption (b), "the strong amalgamation property", is enough to prove m“ #’¢. 1.4.23 Theorem. Assume that n is regular and the G.C.H. holds if n > w. let m satisfy the hypotheses (a)-(c) of Prop. 1.4.22 and suppose also that ‘m is x+-inductive. Then INJ:+(m) # e. In particular, this is true for every free n—class m. .gppgf. Let U E HU:(m) and M e‘m”. Lemma l.3.6(b) guar- antees that there is some embedding of M into U, but to construct a member of INJ:+(m) we will need to obtain proper embeddings. Assumption (b) is ideal for this purpose, but we will need a small lemma. 1.4.24 Lemma. Suppose a is regular, m is a u-class and also n+-inductive, c.= [Ud]a < u+} is a chain of HUfiCM) algebras, and, for all a < n+, Ua < Ua+l° Then C = Lb.€ + INJ: (72;). To prove this lemma, note that C E‘m since m' is n+- inductive. Suppose A‘s B are m gJ < U. Form an amalgam pf" h(Morel) J 4': \\\v/// where h is an isomorphism with h E fa fa(Ma) on Mo' New a'g_A 6 MS“ by assumption (b), and by x- injectivity of U, there is an embedding g: A 4 U such that - 1 on J. It follows that gh = f Ma+l 4 U extends a+l= g es fa and fa+l(Ma+l) nN = NO' Thus (1(2fo = cp: M 4 U is the desired proper embedding, completing the proof of 1.4.23. ‘We have not yet reached our goal which was to prove HUtTS) ¥'¢ for every free n-class 3. To do this we will show that INJK(3) = HUK(3).- This will be accomplished, as it was in Prop. 1.1.9 for 3, by showing that the automor- phisms involved in the homogeneity of U 6 HU:(3) can be chosen inner. Notice that we already know this in the spe- cial case U = K is a countable f.e.p. group by 1.4.3. 1.4.25 Theorem. Let 3 be a free n-class. (g) Suppose A,B,G 6 3<”, A‘s G, B g,G, and m E ISO(A,B). Then there exists V 6 3, with G < V, and some t e V such that G and t generate the HNN extension Gcp in V. 46 (p) Every U E INJK(3) is inner-homogeneous for 3<”, that is, if m 6 ISO(A,B) where A and (n B are 3 subgroups of U, then. m extends to an inner automorphism of U. Hence, (2) INJu(3) = HUK(3). -Proof of (a). Axiom (iv) of a free n-class permits the construction of the HNN extensiOn Gcp in the classical way [15; p. 536]. Let G1 be an a copy of G. Choose N e 3 w is regular and m is u-local and x-inductive, then every m-chain with at least one jump is n-local for m. The proof is similar to the last part of the proof of 1.3.12. 2.1.6 Definition of 0(cl,c2). If (31 and 02 are com- plete chains, then 0(cl,c2) is the set of strictly order- preserving maps from jcl into jc2. 2.1.7 Definition of Embeddings of Chains. Suppose (c1,I1) and (62,12) are complete chains of T-algebras with jci = [J;]d 6 Ii}’ n E 0161,62), and f is an embedding of Lcl into Lb2. We say that f is an neembedding 2; cl into . l l - 2 12 - . I2 2 - _ l - equivalently, f(Jd) g_ “(a) and f(Jd)r1(Jn(a)) - f((Jd) ). . 1 _ 2 (n(a) e 12 is such that n(Jd) — Jn(a))' we say that f is an embedding p§_ c1 into c2 iff f is an n-embedding of 61 into (3.2 for some n e 0(c1,62), and that f is an isomorphism 2;, c1 onto cs.2 iff f is an embedding of cl into (:2 and re ISO(Uol,Uoz). 52 2.1.8 Embeddings of Induced Chains. Using the notation above, suppose E g Lbl is a subalgebra, T] E 0((61)E:02), and fiedcl’02)' We will say that E is a extension pf (extends) T] if, for every (1 6 (Il)E,‘fi(J‘]:-) = n(J€nE)- Notice that if E 6 Cl’ then '3fi extends ‘n' has its usual functional meaning. gplpg_ Definition of a n-Injective Chain for m. J is a n-injective chain for m iff (_l_) .0 E CHM???) and (g) Suppose 6 is an m w is regular, m is a x-class, 1 E m<“, and méx satisfies the S~P.d. If I is any ordered set with 1 g [I] _<_ n, then there exists J 6 INCH? (771) with °J = 1. 2.1.17 Jonsson's Theorem for n-Injective Chains. (G.C.H.) Suppose a 2_w is regular, m is a n-class, m w, m<“ has the s.p.d.. Then, there exists g 6 INCH§”(m), which is unique up to isomorphism of chains, such that if. 016 CH§fi(m), E gJJG is such that CE is an m m be regular. Then, there exist u-injective chains for Jr of arbitrary order types of power .git. Every jump in one of these chains is isomorphic to Ha (see 1.1.6). This is immediate from 2.1.21 with m =.J, our 2nd Existence Theorem 2.1.16, and 2.1.20. Also note the following consequence of the Isomorphism Theorem 2.1.10: If J 6 INCH§”(m), then every order- automorphism of jJ is induced by some automorphism of lJJ. Together with 2.1.22 and 2.1.20, this provides an easy way to see that [Aut fix] = 2". 60 §2.2 An Application of w-Injective Chains of Groups we will apply the construction of u—injective chains to classes of groups m of the following types. 2.2.0 (I) m is a free u-class, where n 2,w is regular, and the infinite cyclic group 2 E m. (II) m is the smallest inductive class containing the subgroups of a fixed countable algebraically closed group G such that the class of f.g. sub- groups of G satisfies the subamalgam property, Nete that such a class m is an w-class, and M e m iff every f.g. subgroup of M is embed- dable in G. (III) m is the class of locally finite groups (an w-class). Note that the free w-class generated by the subgroups of a countable f.e.p. group (see 1.4.21) is a special case of both (I) and (II) by 1.4.19 and 2.1.21. w 0ur application will be the construction of 2 1 non- w isomorphic HU@1(m) groups in each of these three cases, with n = w in case (I). we will first discuss the hypotheses (I) and (II), and then develop the properties of w—injective chains needed for the application. 61 If m is a free n-class, the condition that Z'e m, as in case (1) above, has considerable strength. It allows free amalgamations over 2, and this permits the full use of the HNN and free amalgamation constructions. In partic- ular, the normal basis theorem and the maximal subgroup- tree theorems of §1.2 can be proved for the unique group U“ C HU2(m) where m is a free n-class and 2'6 m. We 2n will not give the details of this, which are mostly obvious I modifications in the proofs of §1.2, making essential use [ of 1.4.25(a) with A = B = 2 for the subgroup-tree theorem. Let us observe that the above groups UK are all simple. , This follows from the following easy facts: (1) Every non- trivial normal subgroup of UK has an element of infinite order; (ii) Every pair of elements of infinite order in UK are conjugate in U“: and (iii) UK is generated by elements of infinite order. The axioms of a free n-class imply that Us is the union of groups which are non-trivial free products, and the facts (i) and (iii) are easily de- duced from this. For (ii), we must use 1.4.25(a) and the axiom 2'6 m, as already noted. These simple observations permit us to test the strength of the axiom 2'6 m because there is a free x-class 3 such that F e HU:(3) is not simple. Using the notation of 1.4.30, let F = F[ZZ] where [22] = 2. Recall that F E HU$(3) where 3 is the smallest free w-class containing 22 which is closed under free amalgamations of its f.g. members. ‘We can show, by induction on the classes In: 62 co n 2_0, (see 1.4.27), that, for every G 6 3* exists in J1 and (x,y) n J1= 1}. Then, (3) For all x E X, Jl = . (p) For all x,y E X there exists t e J such that t'lxt = y. (9) For all aEJl-Ji, we have nX#¢. 2.2.3 Corollary. The Maximality Theorem holds if m is a free n-Class with 2'6 m. 65 Proof of the Corollary. (Refer to 2.2.1) For every jump J of g with J < J1 and H n (Jl-Jl) #91, the three 1 parts of 2.2.2 imply Jl g H since J g H. Hence H is the union of members of ,9 and so H E 51. Proof of 2.2.2(a). Suppose x E X. Let u 6 J1 be any element of infinite order. Since m is a free n-class we can use axiom (iv) of 1.4.20 to find G e m<” such that u E G and there exist c,d e-G such that [c] = [d] = w, u=cd, and c and d generate * in G. Let A E 71¢ g A. Since 2 E 771, we can use l.4.20(iv) to obtain ' G A 9P*(d) 6776“ where d= V . Note that in gp*(a) < u > we have 2.2.4 n A = n A = by the Nermal Form Theorem 1.2.2. By Lemma 2.1.18, there is an embedding 9: gp*(a) 4 J1 with g e 1 on A and g(B) 0J1 = AnJI. It follows from 2.2.4 that n JignAnJignJi=1 since x EX. Since c and x generate < c) * in gp*(a), this proves that g(c) E X(x). Similarly, we have g(d) 6 X(x). This implies u = g(c)g(d) e . Since Jl E INJKW) by 2.1.20, J1 is generated by its elements u of infinite order. This proves that Jl = . 66 Proof of 2.2.2Lb). Let x,y E x and put H = (x,y). Since 3 is a n-local chain for m, there is a subgroup G,) be such that m(x) = y. By 1.4.25(a), the HNN extension ch = g M for some M E 775“. Note that (96 has a maximum jump G and G- = G nJl Britton's Lemma (1.2.3) I.“ implies that and G_ generate *G- in ch : since (x) n Jl = n Jl = 1. Put ng = {Gala 6 16} where jg = [Jam 6 I} and Go = GfiJa for all a E IG as in 2.1.3. For all a E I if J 3 Jo. < J1, define G) Mo. = 2 *Ga 6 775” since 2 6 772: and, if ‘3' . 1311.3:- Jo. < J, define Mo = Ga’ Define G to be the chain whose jumps are jc = [Mama 7! G} u [M]. Thus, M is the maximum jump of G and M‘ = U[Ma[Ga a! G} = . Note that (96 = CG since, for all a 6 I for which Ga 9‘ G, we have G < t’Ga> r16 = Ga again using Britton's Lemma. Let 7‘! map the jumps of ‘96 to the jumps of ,9 which induce them, that is, for all a e I u(Ga) = J where Gd 6’ a Let 1"] be the extension of T) to jc, so that Ema) = GnJa. MG“) for all a E I with Go a! G, and 'fi(M) = J1. Since G 1G is an n—embedding and ,9 is a n-injective chain for m, 1G extends to an fi-embedding f: M 4 U51. Since G n (J-J-) 7! {5, there is some a 6 1G such that fi(Ma) = J; hence f(Ma) S J and since t 6 Ma’ we have f(t) E J and f(t)-1xf(t) = y since f a 1 on G. This proves part (b). 67 Proof of 2.2.2(c). Let a E Jl-J1. Since 9 is a n-local chain for m there is a subgroup G < J1 such that a e G, - . (u, . . . Gr](J-J ) # ¢, and 96 is an m -chain with [396] < u. Let M =p< t) *G where [t] = (1). Thus M 6 771. We define c,n, and B in a manner identical to the proof of part (b), and obtain an fi-embedding f:M “-LW' Since fi(M) = J1, we have f(M) < J1 and f(M) nJI = f(M") = f(< t,GnJ'1'>) = <:f(t),Gr)J1> since f 2 1 on G. Since a 6 J1-J; and = *G, we have (at) n = l, and, applying f, we have nJ; = n (f(M) “‘11) =. <;af(t)>’n<;f(t),Gr)JI> = 1. Hence af(t) E X. Since f(t) 6 J (as in part (b)), we have af(t) 6 and so n X 51¢, proving (c). We next give the lemma pertinent to the proof of 2.2.1 in the case 2.2.0(II). 2.2.5 Lemma. Suppose G is a countable a.c. group and m is the inductive closure of the class of subgroups of G, so that M e‘m iff every f.g. subgroup of M is embed- dable in G. Suppose that g 6 INCHw(M) and J < J1 are jumps of ,9. Put X2 = {x 6 Jl-Jll [x] =2]. Then, the following conditions are satisfied. (p) For all x,y 6 X2, there exists t 6 J such that t-lxt = y, and (9) For all a 6 J -J1, nX2 a! e. 1 68 2.2.6 Corollary. The Maximality Theorem holds if m is the w-class generated by the subgroups of a countable a.c. group whose f.g. subgroups satisfy the s.p.. The proof of this corollary is similar to 2.2.3.. Lemma 2.2.5 will be proved in §3.3. The next lemma proves the Maximality Theorem in the third case of 2.2.0. 2.2.7 Lemma. Let m be the class of locally finite groups. Suppose g 6 INCHw(m) and J < J1 are jumps of 9. For every natural number n 2_2, let Xn = [x E J1-JI[[Jc[= n and n J; = l] and, for all x E Xn, let Xn(x) = [y 6 Xn[<:y,x)»n J; = 1}. Then, for all n 2_2, (a) For all x 6 Xn, Jl = , (p) For all x e Xn and y e Xn(x), there exists t E J such that t‘lxt 2 y, and (2) For all aEJl-JI, an7¥¢. 2.2.8 Corollary. The Maximality Theorem holds if m is the w-class of locally finite groups. The proof of this corollary is similar to 2.2.3. we will note here that the class of finite groups has the s.p.. This can be proved using B.H. Neumann's permu- tational product construction [16]. ‘We will give the details of this in §3.2. Hence, our lst Existence Theorem (2.1.15(b)), together with 2.1.20, imply 69 2.2.9 Proposition. For every countable ordered set I, there exists 3 E INCH§w(£.f. groups) with ”y = I. Every non-trivial member of y is isomorphic to the (unique) countable ULF group Hw e INJw(£.f. groups) (see l.l.6(ii)). The proofs of parts (a) and (c) of 2.2.7 will be de— ferred until §3.2 because they involve special amalgamations of finite groups. H'h".A-ui.“il .7 Proof of 2.2.7(b). Let x e Xn and y e Xn(x). Since o.’.I--m [x] = [y] = 11, there is a finite group G = such that t'lxt = y (see the proof of 1.1.9) and we can assume WLOG that < t) n = 1, otherwise we can form a holomorph to accomplish this. Let 6 = [1,< t>,G}, E = <:x,y>, and u(E) = J1. Thus, IE is an n-embedding of 6E = [1,E} into 9 since E < J1 and ErWJI = 1. Define fi(< t>) = J and fi(G) = J1. By x-injectivity of g, 1E extends to an fi-embedding f of 6 into 9. Hence f(t) 6 J and f(t)-1xf(t) = y as required. In order to use the Maximality and Isomorphism Theorems w for n-injective chains to construct non-isomorphic HUwICm) groups, one more fact is needed. 2.2.10 Inductive Property_for INCHw(M). Suppose J is an m-chain with jJ 7! g! and, for all J 6 jJ, JJE INCHwOR). Then J e INCme). 70 Proof. Refer to 2.1.9. Since a = w, the chain '6, which ‘we are required to embed into J, has only finitely many jumps. Hence, h(jd) < JJ for some J'e jJ and the re- quired embedding f exists because JJ is w-injective for m. It is the failure of this inductive property if n>(g that prevents the direct use of the following method to + construct non-isomorphic HU: (m) groups if m is a free I‘m. KILHTJI i A . t! x-class with 2'6 m. 2.2.11 Definition of n—Initial Chains. J is a n-initial ¢ on. )4.- i we); 71 iff g e INCHKUR), [jg] = ,3, and, for all J 6 jg, gJ e INCHi’W/z). We denote the class of x-initial chains for 77: by INTLKUR). If g 6 INTLKUR), then °9 is an order type of power x+ such that every initial seg- ment of °g has power Sji. Such an order type will be called a n-initial order type. The most familiar example of a u—initial order type is the ordinal u+. 2.2.12 Theorem. Suppose m is an w-class, ‘mnJi= 1, we have a(x) E (x). In the notation of Lemma 2.2.2, this implies that d(x) 6 for all x E X. Thus, by 2.2.2(a), we have (1(2) 6 (z) for all z lying in the free subgroup *< y) for all x e x and y e X(x), and this implies that u(x) = x be- cause if u(x) = x1, then we have both u(xy) = x1a(y) = J and a(xy) = (xy)k for some j,k E 2, which is impos- i X Y sible unless i = j = k = 1. Thus, u(x) = x for all xeX. Since Jl = we have a 1 on J1, and, since Jl is arbitrary, a = 1 . Us? To prove 2.2.18, we first apply the x-local chain prOp- erty of g to obtain G E 775” such that g G < J @ g H where |t| = 0.) because the group e is the HNN exten- sion of (x; with cp(x) = x (here we have used the hypoth- esis that 2 E 7/7 and 1.4.25(a)). H G Let 4= V - Thus, 9P*(d) =M6772<“. Since nJI = l, the Normal Form Theorem (1.2.2) implies that < t) t (G nJi) exists in M. This allows us to define the chain a, the embeddings n and fl, and the fi-embedding f: M -v J1 which extends 16 in a manner iden- tical to the proof of 2.2.2(b) (p. 66, lines 11-24), and we conclude, as there, that f(t) E J. Thus, H 76 2.2.19 a() = = - Since f s l on G, we have f(x) = x, and so e.- = e. 0n the other hand, if u(x) 2’ (x), then t does not commute with a(x) in gp*(d) and 2.2.19 is impossible. This proves 2.2.18. We can use 2.2.17 to obtain some information about the automorphism groups of the groups of 2.2.16, which pos- sess w-initial chains. 2.2.20 Corollary. Suppose m is an w-class of one of the three types of 2.2.0 and J 6 INTLw(m). Put G = LU and, for each J 6 jJ, put A(J) = {a 6 AutG|a(J) = J}. Then, (i) For all J E jJ, A(J) is isomorphic to a subgroup of AutJ, (ii) For all J < Q 6 jJ, A(J) gam), (113;) AutG = IJ{A(J)|J E jJ}, and (i!) lAutG‘ g_2w; so, if the Continuum Hypothesis holds, we have |AutG| = wl = IGi. Proof. Conclusion (i) holds because, by 2.2.17, for all a 6 A(J), a is the unique automorphism B of G such that B a on J; conclusion (ii) is a consequence of the Maximality Theorem; (iii) holds because, for all a 6 AutG, we have u(J) = J for some J 6 jJ (see the proof of 2.2.14); and (iv) follows because, for all J E jJ, we have J 6 HU$(m), IJ} = w, |AutJ| = 2w, and, hence, |AutG| g wlzw = 2m. CHAPTER III EXISTENCE AND SPECIAL PROPERTIES OF n-INJECTIVE CHAINS §3.l Proofs of the Existence and Isomorphism Theorems. Proof of the Isomorphism Theorem 2.1.10, p. 53. we will give only the proof of part (a) since the proof of (b) is an easier application of the same idea. The proof of part (a) is a standard back-and—forth argument, similar in outline to the sketch on p. 19 of the proof of the Isomorphism Lemma 1.3.6. But, since there is more structure in the present context, we will give most of the details. Referring to the hypotheses of part (a), let B =lJB, C =(JJ, and) jB = {JdIa E I}. Note that 6 and J are order-isomorphic since jB and jJ are. We will first prove 3.1.0 If [jal 22, then [B] = x = ]C|. Proof. we have [B|,‘CI g u by simple cardinal arithmetic since 55J E INCH§”(m) (see 2.1.9). To obtain the reverse inequality, note that the class MPB (see 1.3.11) has the 77 78 comparison property and B 6 INCHK(mPB). Hence, by 2.1.13, if |jB| 2_2, mrB has the x-s.a.p.. We can therefore use an argument similar to the proof of 1.4.22 to show that m“PB ¥'¢, which proves 3.1.0. If [3'31 = 1, that is, a = {1,3}, then B,C e INJi‘Km) and, in this case, the Isomorphism Theorem follows from the Isomorphism Lemma 1.3.6. So, we can assume that |j6| 2.2 and, by 3.1.0, Let cf(u) = the cofinality of n = the smallest car- ”i dinal 0 such that n is the union of 0 sets, each of power < u . Beginning with E = E and n = (see the hypoth- o 7‘o eses of 2.1.10(a)) we will construct a chain {EYIY < cf(n)} of m Q Q and note that A then B 3.1.8 1’ a If a E X A If a e X then Ca 2) A For all a E X, Ba 6 We define, for all a E X, 3.1.9 Ad = algM(Ba, Ca ), 3.1.10 A = of M and the chain u = U{C6'a Z 5 6 X2}, FalgM(B;, Ca ) a algM(Ba, A - . KalgM(Ba,Ca) 1f a E X as follows. = LKBBIa 2_B 6 X1} and Ca; and A and Ca 6 a. if a 6 X1--X.2 Ca) if a e Xlnx2 2 - X1, and 3.1.11 ”0 = [Au'a e X}LJ{A;|a e X} and u = the complete chain consisting of all unions and intersections of subchains of m. From 3.1.8 and the s.p., we have, - <1 Aa,Ac E m Notethat a amalgam a‘\\v// a of a; (c) If a E (Xlr1X2)-XE, then A EC! B0. CO (d) If a. 6 :55, then V Ea J The proof in the three cases (a), (b), and (c) is identical because, in these cases, a Z'XB implies that either BarlE = Barns or CarjE = Cafjfi (or both in case (c)) and the conclusions hold in view of 3.1.12 and 3.1.8. In case (d), we have directly that E; = EIWB; = Brno; (by 3.1.6); o o _ - - A in this case, note that Ed < Ea - ErWBa - Ea' We can conclude immediately from the s.p., 3.1.9, and 3.1.12 that 84 II U3 0) 13 CL 3.1.14 For all a 6 X1, AarlUJh) 2. AarwflJt) ll 0 For all a E X From 3.1.10 and 3.1.13, we likewise have OI 9w! 9. . 3.1.15 For all u 6 x1, A‘r1(Lm) For all a 6 x2, A-rlUJE) Taken together, 3.1.14 and 3.1.15 imply 3.1.16 For all a 6 X1, Ba"Bd g_Aa-Aa; and For all a 6 X2, Ca-Ca g_Aa-Au. .Thus, the Conclusion (2) of 3.1.5 holds, and so does con- clusion (3) if we define f = HJE' Also note that 3.1.14 and 3.1.15 imply that, for all a e x, (A;,Au) is a jump of u. we must yet show that.conc1usion (1) holds; namely, that every jump of u is of the form (A;,Aa) for some a e X. First note that 3.1.17 For all a e x, A; = LKABIG > B e X}, which follows from the facts that, for all a 6 X1, Ba =|J{BB‘G‘> B 6 X1} and the analogous fact for Ca' Also note that M = U{Aa|a. E X} since M = (4). Now, if X is well ordered, conclusion (1) is an immediate con- sequence of 3.1.17. So, we must attend to the case in which assumption (vi) holds (the descendance condition (d.c.) - see 2.1.14) and X is not necessarily well ordered. From the d.c. we have 85 3.1.18 For all a E X, Ac = FWAéIG < B E X}, which is the dual of 3.1.17. Suppose S is a subset of X and I = {Acid 6 S}. We will prove 3.1.19 If I does not have a largest member,_then either UT = A; for some T e x or UT 4- rflAYlY 6 8+} where S = {Y E X|Y > a for all 0653}. This will show that v has no jumps besides (A;,Aa), a E X: Suppose (J_,J) is a jump of a such that J- # A; for all a E X; then we also have J- #‘Aa for all a E X for, otherwise, 3.1.18 is contradicted. Hence J— g'qo (see 3.1.11) and so 3.1.20 J- =IJU where I = {Aa|a E X and Au < Jr} because m is the completion of mo. Since 3.1.20 contra- dicts 3.1.19, it will indeed suffice to prove 3.1.19. To prove 3.1.19, assume I = {Aala 6 S} has no largest member and put S1 = xlrws and $2 = erjS. We can assume WLOG that 3.1.21 LU = LKAle E 81} and, hence, Sl has no largest member. We define c: l "' U£BYIY 6 $1}: N< 2 = {p e lep le for some Y 6 81}, 86 V V = U{CH)U 6 X2}, + S1 = {a 6 X1}a > Y for all Y 6 SI}, V+ V X2 = {B E X2|B > Y for all Y 6 X2} {B E X2|B Y for all Y e 31}, and \/ D II + 2 {B E X2|a < B < Y for all a 6 S1 and Y e 31}; and we note that _3_-_l_-_2_2_ U? = . ‘We will check that 3.1.19 holds by considering the following three sets of exhaustive cases. 3.1.23 (1) A2 d and (2) A2 # ¢. 3.1.24 (i) U = B; for some .a 6 X1. In this case, a is the smallest member of S (ii) Otherwise. In this case, since U cannot be an upper member of a jump of m by 3.1.21, we have .1. U = niBYIv 6 81}. 3.1.25 (a) V = C5 for some B 6 X2. This is the case iff B is the largest member V of X2. (b) V = CB for some B 6 X2. In this case, B is the smallest member of X3. (c) Otherwise. v In this case, we have V = F[Culp 6 X3}. The following table shows the status of LU sible case, and 3.1.19 is 87 readily checked from it. in every pos— (l): (2): A2 7! ga’ A2 = ¢, (a): (b):_ (C)t V = C£3 = C6 OtherWlse (1): UI _ A_ w LIT Lxr U = Ba a (ii): (JT = Other- _ _ the that, because of 3.1.21, we have SI 3.8+ and Az‘g V+ x2 g_s+, and so 3.1.19 does follow in all cases. We will discuss only one case here since they are all quite similar. V largest member of X2, that A Y = In the case (2)(a) we have and A2 #’¢. (see 3.1.7-9). (I) B has an immediate successor, T, V: For all Y 6 A2, in X2; CB: E = note we consider two cases: evidently 'r 6 A2, CB = C1" and hence U? = = = A,r (see 3.1.10), and (II) 2 has no smallest member; in this case, C6 = Fficle 6 A2} = Fficle E A2} since C6 is not 88 the lower member of any jump of a as in the previous case, and so UT = = = fl{A;|Y 6 A2} by the d.c.. This completes our proof of Lemma 3.1.5. The set-theoretic component of the construction is given by 3.1.26 Lemma. Suppose X 2_w is regular and that the G.C.H. holds if x > m. Let (s e CHKUIZO‘) and let x be the class of all amalgams U58 US a = \\\\/// such that (l) E $,LE is such that GE is an m<“-chain and m e CHn(m<”) satisfies the hypotheses of Lemma 3.1.5 with f = 13; and (2) XlLJX2 = X's I where I is a fgxed totally ordered set of power n. Then, there are at most a s-isomorphism classes of X amalgams where the amalgams a1,ab e X are §:isomorphic . _ o __Q _ . . lff E1 - 32, $1- $2 — X1, and there IS a function g of III 41 onto ab such that g 1 on L5 and g a some IXi—lsomorphlsm of 81 onto 32 on (J31. grggfi. we will contrast this computation with that used in Jonsson's Theorem. First we will give the pertinent definitions and facts from cardinal arithmetic. Suppose x and o are infinite cardinals and U is a set of power a. 89 w H N \l "U C'. V II the power set of U; P w and x 2,w is regular, then 3.1.28 a = 2<“ = “<“. In proving Jensson's Theorem we must count the number B C of C-isomorphism classes of m m: if a = w, the hypothesis that m a such that g is a 1x -embedding of E into KY and g E l on E. 1 Proof. Lemma 3.1.26 guarantees that, for each Y < u, there Us Uta are at most n amalgams \\\/// up to ca—isomorphism E , which require the existence of an injection 9 as in 3.1.32 into some sy, Y > a; and, if Y > a and Us USY d = \\\/// with m and E as above, then we can use E Lemma 3.1.5 Wlth f = IE to obtain the chain a = sy+l with IY+l = XluIY g_I (where Xl==°m) such that 3.1.31 92 is satisfied by 3.1.5(2) and such that some lx —injection, 1 g, of m into GY+1 ex1sts which extends 1E. Since there are n2 = n non-limit steps available, all of the required injections can be built into the chains. At limit ordinals l < x, the chain. ax is defined as follows. ET}- 3.1.34 1x = U{IY|Y < l} and, for all 1 e 11’ 1 _ l 3 J1 —IJ{JtlY < A and 1 E Iy} and : (J:)- = LK(JZ)-IY < l and 1 6 IY}: and 5 51 = the completion of the chain L X 1 ' {J1} (J1) It E IA}. The property 3.1.31 implies that, for all Y < l and Y_ Y" K- 1" ' 16 Iy, J1. (J1) gJt (J1) . Thus. th ,Y} and all the injections g built into the chains at previous = {J:|1 E I steps still exist into 61' To define J 6 INCH§”(m) ‘with °J = I, note that U{IY[Y < n} = I since the chains m have arbitrary order types xl-S I, |Xl| < x. New, the definition of J is carried out exactly as in 3.1.34 with l = n and J = S u' To check that J is n-injective, suppose lJB lJfl 3.1.35 \\\/// is an amalgam as in 3.1.32 with J in E place of Ga. 93 Since E e m for all m22. J1 a Hm (see 2.2.9, p. 69) is generated by its elements of order n [2: p. 305]; so, to prove 3.2.3, it suffices to prove 3.2.4 For every z 6 J1 with |z| = m, we have 2 E . .,h To Prove 3.2.4, let z E Jl with [2] = m, and let G = ® be such that [a] = lb] =m and z = a+b. By 2.1.18 and w-injectivity of g, there is an embedding l l on <2) and f(G)nJ1= <:z>»nJi. Hence, f(a),f(b) E Xm, proving 3.2.4 and 3.2.3. f: G 4 J such that f Now suppose x e Xn. In View of 3.2.3, to prove 2.2.7(a), namely J1==<:Xn(x)>, we need only prove 3.2.5 Xm< for some m22. Proof of 3.2.5. Let m be a prime dividing n and suppose w e xm. We wish to show that w e . Since xexn(x), we can assume WLOG 3.2.6 w E . Let H = ® be such that [a] = [b] = n and H 3.2.7 the amalgam a = exists, where w = (ab)n/m. By the s.p., let G = <¢7> be a finite group such that < do) fld = do for every subamalgam do of a. By Lemma 2.1.18 there is an embedding g of G into Jl such that 96 3.2.8 951 on and g(G)nJl=nJl From 3.2.6 and 3.2.7 we see that 3.2.9 V and \/ are subamalgams l l . of a, 5.. and, by the s.p., we have 3.2.10 n = and n = . Now, 3.2.8 and 3.2.10 imply g()nJ'1' = g() n nJI = (x) nJ; = 1 since g u l on and “~— x e Xn' Hence, g(a) e Xn(x), and, similarly, we have g(b) E Xn(x). Since w = g(w) 6 g(H) = , 3.2.5 follows. Proof of 2.2.7(c), p. 68. Let a 6 J1-JI. Since parts (a) and (b) of 2.2.7 have already been proved for all n, if we now prove 3.2.11 nXm 7! d for some m 2 2, it will follow immediately that = Jl proving (c). Proof of 3.2.11. Put [a] =n and nJi=. Let G be the group with presentation (a,b: an = bn = 1, ba = b, am = bm). Using 2.1.18, we obtain 9: G -o J1 such that ga l on and g(G)nJi=. Let us identify G with g(G). Let M= where at=b, bt=a, and [t] = 2. Note that © exists in M. Using the 97 method on p. 66, we obtain f: M 4 J such that f a l l on G and f(t) 6 J. Since GnJI = (am) = (hm), we have ab-1 6 an, proving 3.2.11. Proof of the Theorem on Uniqueness of Automorphisms 2.2.17, p. 74, for the class of Locally Finite Groups. Suppose a. 6 Aut(Ug) and, for some J 6 jg, a s l on J. We will "5' prove 3.2.12 For every element x e UW of order 2, a(x) = x. This will imply that a = lUfi since Ug is generated by 1.; -3: its elements of order 2. Proof of 3.2.12. Suppose x e Ug has order 2. Since we wish to show that u(x) = x we can assume WLOG that x 2’ J and that 3.2.13 x eQ-Q- where J® where G H t a! 1 has finite order, and let a = V . Let M = be a finite group such that < go) ['14 = do for all subamalgams d0 of 4. Since u(x) f, t and u(x) do not commute in gp*(d), and we can assume WLOG that 98 3.2.15 t and u(x) do not commute in M because gp*(a) has a free subgroup of finite index [6: pp. 227-228] and, hence, M can be replaced by a larger finite homomorphic image of 99*(d) in which t and u(x) do not commute. Recalling that jg = [Jala E I}, we define, as in the _ r proof on p. 66, for each a E IG = [a E I|Gf1(Jd-Jd) ¥ ¢}, g GrWQ- <1t> Ga = GrWJa and Ma = < t,Ga:>. Since \\\\\////' ls l a subamalgam of a, the subamalgam property of M = <¢7> implies (600-, t) no = on o" and, also, 3.2.16 For all a 6 I such that Ga # G, we have G (Ga,t>nG = Ga. Using 3.2.16, the proof on p. 66 can be copied (lines ll-27; 3.2.16 is used in place of Britton's Lemma) to obtain the embedding f: M 4 0 such that f s l on G and f(t) E J. New, x = f(x) commutes with f(t); but, u(x) = f(d(x)) does not commute with f(t) by 3.2.15. Since f(t) E J, we have a(f(t)) = f(t) and d() = . This contradiction proves 3.2.12. 93.3 w-Injective Chains of Algebraically Closed Groups. In this section we will give proofs for Lemma 2.2.5 on p. 67 which establishes the Maximality Theorem for classes obtained from a.c. groups and for the Uniqueness of 99 Automorphisms Theorem 2.2.17 on p. 74 for these same classes. For both proofs we will assume the hypotheses of Lemma 2.2.5, namely that 3.3.0 G is a non-trivial countable a.c. group,‘ m is the inductive closure of the class of subgroups of G, and g e INCHw(m) with [jgl 2_2. Notice that it is not at all apparent which a.c. grOups G are capable of satisfying this hypothesis. The existence of g implies by 2.1.13 that m is either a finite dihedral group or z IkZ (the infinite dihedral group), 2 2 H has a solvable word problem and, hence, so does Hm = , the HNN extension with :96 ISO(,). Hence Hm e m and the proof of 2.2.2(b) on p. 66 is applicable with M = Hw. we obtain, as there, an embedding f: Hw 4 J1 such that f a l on H and f(t) E J and the desired con- clusion follows. Proof of 2.2.5(c). Let H =<*<:t> where [t] = 2. Thus, H has a solvable word problem and H E m. By w- injectivity of g, there is an embedding f: H 4 J1 such 101 that f a l on , f(t) 6 J, and, for all Q E jg such that JgQng, f(H)nQ=<flQ, f(t)>. The exact details are similar to those on p. 66 and depend, of course, on the normal form theorem for H. Thus, f(H)nJI=<nJI,f(t)> and, since a (J1, we have a-lf(t)a Z Ji. Hence a-lf(t)a 6 X2r1, completing the proof. m1 Proof of 2.2.17. The proof is the same as that on pp. 97 and 98 of this same theorem for the class of locally finite groups with two differences: (1) we do not need to demand that Gr1(J-J-) # ¢; this was done only as a matter of 5" convenience on p. 66; hence, we can put G = , a group with solvable word problem; and (2) we put M = 9P*(d) e 77: by 3.3.2(c). [’0] [ 1] [ 2] [ 3] [ 4] [7 J [ 8] [ 9] [10] [11] [12] BIBLIOGRAPHY Baumslag, G., Karrass, A., and Solitar, D., Torsion- free groups and amalgamated products, Proc. Amer. Math. Soc. 24 (1970), 688-690. Bell, J. and Slomson, A., Models and Ultraproducts, 1 Amsterdam: North-Holland, 1971. E Hall, P., Some constructions for locally finite grdups, J. London Math. Soc. 34 (1959), 305-319. 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