-. . an“. _.__..-- “a R“ (IN MARGINAL SUBGROUPS AND THEIR GENERALIZATIONSI Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY TOMMY KAY TEAGUE 1971 This is to certify that the thesis entitled ON MARGINAL SUEiROUPS AND THEIR GENERALIZATIONS presented by Tommy Kay Teague has been accepted towards fulfillment of the requirements for Ph.D . degree in Mathematics MJZ M154 Major professor Date 92"] as", /9]/ 0-169 81‘0‘ gar. “he 01% d6: ABSTRACT 0N MARGINAL SUBEROUPS AND THEIR GWEALIZATIONS By Tommy Kay Teague In this paper several problems concerning marginal subgroups are investigated. in elenent a of G lies in the marginal subgroup ¢'(G) corresponding to the word ¢ if and only if ¢(gl, ... , gn) = “81. ... , agi, ... , 3n) for all choices of g1, ... , 3n inG and i = 1, 2, ... , n. Let A0 ‘-'-' l, and define Ami-l to be the complete inverse image of ¢"(G/Aa). If a. is a limit ordinal, define Au = U {A88 3 < a}. A group G is called ¢-nilpotent of class n 11‘ there is a positive integer n such that and # A.n = G, and G is called ¢-hypercentra1 if there is an ordinal B such that AB = G. In Chapter 2 some of the basic properties of ¢-hwpercentral groups are developed. The following theoruns are proved: ghggren. Let G0 = G, and inductively let Gall be the subgroup generated by elements of the form ¢(g1, , gn)'1¢(glh1, , gnhn), where each g1 c G, hi e G“. Define G“ = (KGB: B < a} for a a limit ordinal. Then G is ¢—nilpotent of class n if and only if 6"“1 a! G“ = 1. m. Acme G is ¢-hypercentral. Then ¢(G) has a descending hypercentral series. gheorun. Let ).(y1, ... , ym), 6(x1, ... , x“) be two words such that 6(6) Q “6) for all G. Set ¢(y1, , ym, x1, , xn) =. Tomnw Kay Teague (Marl, ... , ym), 9(1'1, ... , xnf]. If G is a group such that HG) is nilpotent of class c, then G is ¢—nilpotent of class no greater than c. Theorem. Let dn be the n-th derived word. For arw G, G is dn-nilpotent if and only if dn_1(G) is nilpotent. In Chapter 3, two generalizations of the marginal subgroup of G are considered--one in G and the other in Aut(G). Define ¢°(G) = {a c G: ¢(gl, ... , gi, ... , gn) = ¢(g1, ... , gn) for all choices of g1, ... , gn in G and i = l, 2, ... , n} to be the c-marginal subgroup of G. By substituting a c Aut(G) for a c G in the definition of ¢°(G), it is possible to define the automargin 3(G) _C_:_ Aut(G). Theorem. ¢°(G) = {a c G: ¢(gl, ... , agi, ... , gn) = “81. ... , gia, ... , gn) for all choices of g1, ... , gn in G arrii=1, 2, , &. Theorem. Define Yn = [3:1, ... , xn]. Then y;(G) = Zn(G) for n _>_ 1. Ihgrem. Let ¢ be any outer commutator word. Then a(G) = {a c Aut(G): [G, a] _C__ ¢'(G)}. In particular, Vn(G) = {a c Aut(G): x0. x mod zn_1(e) for all x a G} . Marginal subgroups for outer commutator words were completely characterized by R. F. Turner-Smith in 1964. In Chapter 4 the marginal subgroup for another type of commutator, the Engel word e2 = [x, y, y], is determined. W. cam) = {a e G: [x, y, a][:a, y, X] = l for all x, y c G}. 92mm- eye) n CG(G'> = 22(6). m. For m a c eye), [a, a, 613 = [.3, e, c] = 1. Theorem. 22(G) (_:_ e§(G) ; 23(G), and e§(G)/ZZ(G) is an elementary Abelian 3-group of central automorphisms on G' . Tomy Kay Teague m. If Z(G') flue) has no elements of order 3, or if G' has no proper subgroup of finite index, then e;(G) = 22(G). m. If [G: 5(a)] = m is finite, then e2(G) is finite with order which divides a power of m. m. If G is locally residually finite and e2 is finite- valued on G, then e2(G) is finite. 121m. The preceding two theoruns also hold for the Engel word 03 = [X. y. y. y]- ON MARGINAL SUEiROUPS AND THEIR GENERALIZATIONS By Tomy Kay Teague A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTCB OF PHILOSOPHY Department of Mathenatics 1971 It ."‘j.| Richard of the II preparat ACKNCWIEDGMENT The author wishes to GXprOSS his gratitude to Professor Richard E. Phillips for his many suggestions, his careful reading of the manuscript and, above all, his encouragement throughout the preparation of this thesis. ii INTRODUCTION CHAPTER I. CHAPTER II. CHAPTER III. CHAPTER IV. BIBLICBRAPHY Tm OF WNTENTS PRELIMINARIES immcmm GROUPS . . MARGINAL AUTOMORPIUSMS . THERIGELMARGIN . . . . . iii 18 26 1+3 INTRODUCTION The concept of a marginal subgroup for a word ¢ was introduced in 1940 by P. Hall [4]. It is known that the marginal subgroup of a group for the word [x, y] is the center of the group. By analogy with the ascending central series of a group, we may define its ascending ¢-series. Further, we may generalize the marginal subgroup itself by considering elments in the group which do not change the value of the word when they conjugate any of its variables. This in turn leads us to consider automorphisms which do not affect the value of the word when they are applied to arv of its variables. R. F. Turner-Smith [21: page 328] has completely characterized the marginal subgroup for an outer cmutator word. We conclude this paper with a characterization of the marginal subgroup for the Eh'igel word of length two, which is not an outer commutator word. For the sake of coupleteness, Chapter I contains some definitions ani theoruns essential to the reminder of the paper. In Chapter II we develop some of the basic properties of ¢- hypercentral groups. Marry of the usual theorems for hypercentral groups are true in this new context. We are also able to define a lower ¢-series for a group. We give a complete characterization of dn-nilpotence for any derived word :1n and necessary conditions for some other words. Some possible ways to generalize the normalizer of a subgroup are also suggested . 2 Chapter III offers two generalizations of the marginal subgroup-- the c-marginal subgroup and the automargin of a group. An alternate characterization of the c-marginal subgroup is given, and this subgroup is computed for star nilpotent word. The automargin is shown to be minimal for outer commutator words. In particular, the automargin of a group G for a nilpotent word is shown to be the group of n-normal automorphisms on G for some n. In Chapter IV we consider the marginal subgroup E for the Engel word of length two. A complete characterization of E is given, and several interesting properties are brought to light. The relationship of the subgroup of right Rigel elenents of length two to the metabolian margin is considered. Some results concerning the relative size of the verbal and marginal subgroups of G for the Engel word of length two are also presented . CHAPTER I PREMNARIES W w W. Let G be a group. The symbol 1 is used interchangeably for the identity in G and for the unit group. By a QC. (3 < G) (a <1G) (H c Char(G)) we mean that H is a subgroup of G (a proper subgroup of G) (a normal subgroup of G) (a characteristic subgroup of G). If H _C__ G, [Gas] is the index of H in G, 06(3) is the centralizer of H in G, and NG(H) is the normalizer of H in G. If S is a subset of G, then <3 is the subgroup of G generated by the eluents of S. The order of an eluent g of G is written o(g), and 0(8) denotes the cardinality of the set S. The infinite cyclic group is represented by J, the cyclic group of order n by Jn, the symmetric group on n sym- bols by Sn’ and the alternating group on n symbols by An. If G‘ is a group for each a in some indexing set A, then the (unrestricted) direct product of {Ga} a c A} is denoted by "{G‘: a e A} and the direct Sumhytfilal a sh}. ByG§ 3 (egg H) wemeanG is isomorphic to (a subgroup of) B. By a class of groups we mean a class containing the unit group as well as all isomorphic copies of amr manber of the class. Let 2 be a class of groups. Then: (a) $2 is the class of groups which are subgroups of )3 groups. (b) Q}: is the class of groups which are quotients of 2 groups. (c) E: is the class of groups which are extensions of 2 groups 3 by 2 groups. (d) L2 is the class of groups in which every finitely generated subgroup is a 2 group. (e) DP): is the class of groups which are direct products of 2 groups. If p;{s, q, E, L, or}, and if P22 2, we say 2 is P closed. The center of G is CG(G), denoted by Z(G). Let 20(G) = l, 21(G) = Z(G). The ascerriing central series of G is defined recursively by Zw1(G)/ZQ(G) = Z(G/Za(G)) for all ordinals a, and Za(G) = UfZB(G): e < a} for all limit ordinals 0.. If there is a least finite ordinal n such that Zn(G) = G, then G is nilpotent of class n. If ZB(G) = G for some ordinal B, G is a ZA group. IfG has an ascending normal series 13 GO . RE, KQG, then[h, K]=<[h, k]: heH, ch>. Ifxi eG, lsign, n23, wedefineEJcl, ... , xn] recursivelytobe [[xl, ... , xn-l]’ xn]. Similarly, if X1(;_—._G, l S. i 5 n, nz 3, we define [$1, ... , In] = [[Xl, ... , £51], In]. A subgroup A of Aut(G) stabilizes the normal series G = G0 9 G1t> ...DGn= lofG iIGQ=G1am[G1, figsfil, Ogisn- 1. We shall make frequent use of the fact that if Gi <1 G for each i, then A is nilpotent of class 5 n — 1 (see [7]). Also, for A I; Aut(G), F(A) = {g c G: [g, a] = 1}. Define [x, 1y] = [x, y] and Ex, my] recursively to be [[x, (n - Dy], y]. An eleuent g c G is a left Engel elauent if to 5 each x e G there is an integer n depending on x such that [x, ng] = 1. Similarly, g c G is a right Engel elment if to each x c G there is an integer n depending on x such that [g, nx] = 1. If every elenent of G is a left mnel element, then G is called an Engel group. If H, K C; G, define [11, 1x] = [3, x] and [3, mg recursively to be [[H, (n - 1m], K]. A group G is residually finite if for each 1 1 x e G there is a normal Nx <1 G with x ¢ Nx such that G/Nx is finite. An elenent a e G has infinite height in G if the equation it“ = a has a solution in G for each integer n. A group G is complete if each a e G has infinite height. A group G is ‘éernikov complete if for each integer n G is generated by the n-th powers of all its eluents. A group G satisfies the maximum (minimum) condition if each proper according (descending) chain of subgroups of G is finite. By [17: Theoran V1.7.b] every group has a unique maximum locally nilpotent normal subgroup. We call this subgroup the Hirs ch-Plotkin radical of G. A word is an elenent of the countably generated free group < x1, x2, ... >. A law in a group G is a word such that every substi- tution of slanents from G for the variables of the word yields the identity of G. If S is a set of words, then the variety determined by S is the class of all groups G such that the elenents of S are laws in G. For am word ¢ we denote by ¢(G) the verbal subgroup of the group G generated by all the values of ¢ obtained by substituting elenents from G for the variables of ¢. The associated marginal subgroup ¢"(G) of G consists of all a c G such that ¢(g1, ... , 3n) = “g1. ... , agi, ... , gn) for every g1 e G, i = l, 2, ... , n. We also refer to ¢"‘(G) as the ¢.nargin of G. 6 The word Vl = y1(x) = x in one variable is an outer comnutator word of weight w(y1) = 1. If 6 = 6(x1, ... , xn) and l 3 Myl, ... , ym) are defined outer commutator words such that w(e) = n and w“) = m, then ¢ = ¢(x1. . 25,”) = [90:1, . x“). "(xnv-v , agnmfl is an outer comnutator word of weight w(¢) = m + n. We write ¢ = [9, l]. Particular mmples of outer commutator words we consider are the derived (or solvable) words, defined by do = x, d = n [d104, dud], and the nilpotent (or lower central) words, defined by Y1 = ’9 Yul-l = [Yn’ Y1]- We define en = en(x, y) = [x, rw] to be the keel word of length n. For n > 1, we note that °n is not an outer commutator word. Most of the itens referred to in this section are discussed in detail in [lo], [13]. [17]. [18] or [21]. m. We include here some known results on marginal subgroups. For the proofs, see P. Hall [5] or P. W. Stroud [20]. IMLL For arqr group G and word ¢, (a) ¢(G) is fully invariant in G and ¢‘(G) c Char(G). (b) ¢(¢‘(G)) = 1. (o) if x/¢‘(G) = z(G/¢*(G)), then [x, ¢(G)] = 1. In particular, U76). ¢(G)] 3 lo (d) if H _C;_ G such that G = MVG), then ¢"(H) = Hfl ¢"(G) and NW = ¢(H)- 11m 1...... If {Gas a c A} is a set of groups, 95 a word, then ¢*(11{Gaz a e A}) = u{¢"(Ga)x a a A} . mm. Let a = 9(x1, ... , xm) and x = Myl. ... , yn) be two words ard let ¢ = ¢(x1, , xm, yl, , yn) :- 7 [90:1, ... , 35“), Myl, ... , yn)]. Then in am group G: (a) ¢(G) = [6(6), x(G)]. (b) if U = CG(9(G)), v = cG(l(G)). L/U = l*(G/U). M/V = 9*(G/V), then ¢"'(G) = Lfl M. m M. y;(G) == Zn_1(G) for any group G and n 2 l. m 1.5. If e and x are words such that MG) _C_ e(G) for all groups G, then e“(G) Q l“(G) for all groups G. We shall also need the following theorans. For the proofs see [17] or [18]. m b.6- (N/C Theorem) If H _C_ G, then NG(H)/CG(H)C,; Aut(H). ‘ Theorgg L1. (3 Subgroups Leanna) If L, M, and N are subgroups of a group G, then [L, M, N]C' _C_ ([L, N, HIM, N, L])G. Ingran Lg. (Levi's Theorem) If e2 is a law in a group G, then G is nilpotent of class at most three and y3(G) has exponent dividing three. CHAPTER II ¢-mmcmm1. GROUPS In this chapter we shall eXplore some generalizations of groups with transfinite ascending and descending central series. Unless we state otherwise, ¢ = ¢(x1, ... , xn) is an arbitrary word in n variables. 9331mm L1. (a) An ascending invariant series where ~ A0 = l, Adi-1M0. C ¢'(G/Au) for each ordinal c, and Au = U {A83 8 < a} for a a limit ordinal, is called an m Eagles for G. Inli Eli-m 1" 'v‘lanr J- ' -‘. l l (b) The p.229; Q-sgges for G is the ascending ¢-series where Adl/Aa = ¢‘(G/Aa) for each ordinal c. eorau 3,3. Let A0 = 1 <1 A1 4 ... be an ascendirg ¢-series for G, and let G0 = 1 <1 G1 4 ... be the upper ¢-series for G. Then Au ; Go. for each ordinal a. m. We induct on c. Certainly Au _(_:_ Ga for c = o, 1. Thus assume AB Q GB for all 1 5 B < a. If a is a limit ordinal, then A“ = UfAB: B < a}; U{GB: B < a} = Ga by the irrluction hypothesis. Now assme a-l exists. Suppose a c Au so that “(1-1 s ¢'(G/Aa_1). Then we see that ¢(g1, ... , agi, ... , shun]. al- ¢(g1, ... , gnu“ _1 for every 1 and g1, ... , gn in G. Since A(1» 1 Q G'0.» “81: ... 9 381’ ... a Sr,” “.1 "¢(81. ... , gn)Ga_ 1. Hence a c Go. lby the induction hypothesis, we must also have that and the theoran follows. Definition £1.3- Suppose G0 = l a G1 d ... is the upper ¢-series for G. If G = Ga for some ordinal 0., then G is Q-Mercentral. If 8 9 G = Gn 1 Gnu1 for some finite ordinal n, then G is Wm 93 glass 1;. We note by Theoran 2.2 that a group G is ¢-hypercentral if and only if every properly ascending ¢-series for G reaches G. Ihggrem 2.43. Subgroups and homomorphic images of a ¢-hyper- central group G have upper ¢-series of length no greater than that of the upper ¢-series for G. 23922. Let G0 = 1 <1 G1 <1 ... <1 Ga = G be the upper ¢-series for G, and let H g; G. we claim that H n Gal/H 0 Ga _C_ ¢‘(H/Ga n H) for each ordinal (1. Suppose a c 60+]. 0 H\ Gan H. Then a c Gui-l and “"1” , ahi, , hn)¢(h1, , hi, , hn)'1 cGan H for every 1 and hl’ ... , 1'51 in H. Hence “(Gan H) c W(H/Ga F) H) and Ho: l. 1112223211. g,_6_. (Rhemtulla) 'For any G, ¢(H A G) is the smallest K <1 G, x g H such that H/K 5:; ¢‘(G/K). Also, ¢(H) Q ¢(H A G) _C_ H n¢(G). Ifih'fl'WVK-GLY‘ 10 Theoren 2.6 suggests a way to describe descending ¢-hypercentral series as well as ascending ones. W 241. (a) A descending invariant series A0 = G 9 A1 9 ... of a group G where Au/Amlg ¢‘(G/Ac,.l) for each ordiml 0., and Au =n {A83 5 < a} for c a limit ordinal, is called a W £2212 for G- (b) The m B35193; for'G is the descending ¢-series G0 = G > 61> ... where Gui-l = ¢(G°‘ A G) for each ordinal a, and 63 II 0 {G83 3 < c} for c. a limit ordinal. The proofs of Theorans 2.8-2.11 are identical to the proofs of the corresponding theorans for central series and will be omitted. The statments thuselves are included only for the sake of completeness. m gag. It Go a G1 c is the lower ¢-series for G and A0 = G > A1» ... is a descendirg ¢-series for G, then G"L 9; AOL for each ordinal a. W _2_,_2. A group G is ¢-nilpotent of class 11 if and only if G“‘1 1 Gn = 1. 111m w. Suppose G/H is ¢-hypercentral, where H C; W (G). Then G is ¢-hvpercentral. MAE. IfG is ¢-Wpercentral andl< H<1G, then H n¢"'(G) > 1. w 241.2. Assume G is ¢-nilpotent of class n. Then ¢(G) is nilpotent of class less than n. m. For convenience, we write the upper ¢-series as G = G() :> G1 > 1> Gn = 1, where (ii/G1+1 =¢*(G/G1+1). Define D1(G) = y1(¢(G)) for i 2 1. We claim that D1(G) _C_ G, for 1 5 i 5 n. Since G/G1 = ¢*(G/Gl), we have that ¢(G)G1/G1 = ¢(G/Gl) = 1 and thus Wt fir -xu 11 that D1(G) = ¢(G) E G Assume then that 01(G) (_:_ G1L for i ,>_ 1. 1. Hence D1(G/G1+1) = D1(G)Gfi1/G#1C;_ G1/Gfi1 = ¢"'(G/G1,1) and D1+1(G) = [D1(G), ¢(G)] ;_ Git-1‘ Thus D1(G) Q; 61 for each 1 5 i 5 n. In particular, Dn(G) Q; Gn = 1 an! ¢(G) is nilpotent of class less than n. M 3‘13. Suppose G has upper ¢-series G0 = l 4 Gl a" ... <1 G0 = G. Then [God-1‘ ¢(G)] (_._—__ Go. so that the group A g ¢(G)Z(G)/Z(G) of automorphisms on G stabilizes the upper ¢-series for G. By a result of Hall and Hartley [7: Theorem 11], a and consequently ¢(G) have a desceniing hypercentral series. m 3.111. The converse to Theorem 2.12 is not true. Let G = 83. Since 73%) = 2(G) = 1 by Theorem 1.1+, G is not y nilpotent. 2. However, y2(G) = G' has order 3 and is certainly nilpotent. m £45. If e and l are two words such that 9(G) t; 1(G) for all groups G, then a A-hypercentral group is also 9-hypercentral. m. Letl<1A1<1 ... .(G>] for i Z 1. By Wpothesis, Sc(G) = l, Sc_1(G) 1 1. We induct on c. 12 For c = l, we have that l = 81(G) = EKG), A(G)] = EKG), 9(6)] = ¢(G). This implies ¢*(G) = G. Hence G is ¢-nilpotent of class 1. Now assume that for every group G such that MG) is nilpotent of class _<_ c, G is ¢-nilpotent of class 5 c. Let MG) be nilpotent of class c0-1 for some G. Then Sd1(G) = [Sc(G), 1(6)] = 1, Sc(G) 7‘ 1. By Theorem 1.3(b), Sc(G) ; CG().(G)) n CG(6(G)) (_:__ ¢"'(G). Let G = G/¢"‘(G). Then S ca?) = 32257 = '1'. By the induction hypothesis, s—r G is ¢-nilpotent of class 5 c. By the arguments used in the proof of I Theorem 2.10, G is ¢onilpotent of class 5 ed. m 5.1.8; Let G = 53 and recall d2 = [d1, d1]. Since d1(G) = G' is nilpotent, G is dz-nilpotent by Theoran 2.17 but not {gr-Ir; dl-nilpotent. Hence the converse of Theorem 2.15 does not hold. M212 .2412. Let 6 = [¢, ¢] be a word in 2n variables. Note that we get the result of Theoran 2.15 for 6 by Theorems 2.12 ard 2.17. That is, if G is ¢-nilpotent, then ¢(G) is nilpotent by Theoran 2.12. By Theoran 2.17, G is b-nilpotent. The advantage of Theoran 2.17 in general is that ¢(G) nilpotent is a weaker condition than G ¢-nilpotent, as was shown in Example 2.11+. Ebcample 2.429. The hypothesis that 6(G) 9; ).(G) for all G in Theoran 2.17 is essential. Let H = 83. Note Y3 = [y2, VI] and y2(G) Q; y1(G) for all G. Then H is not ya-nilpotent although y2(H) = H' is nilpotent. m 232),. For any nilpotent word Yn define ¢n = Eva, yn]. Then yn(G) is nilpotent if and only if G is ¢n-nilpotent. ' 23292. The necessity follows immediately from Theorem 2.17. Now assume that G is ¢n-nilpotent. We seek another characterization of ¢;(G). By Theorem 1.3(b) and Theorem 1.4, uni,” t~_'.‘ 13 ¢:,(G)/CG(Yn(G)) = v;(G/Cg(vn(G))) = Zn_1(G/Cg(yn(G))). Hence ¢';,(G) = {a e G: D. 82. 53. . an] e CG(vn(G))n yum) = Z(Yn(G)) for all g2, ... , gn in G} = {a c G: [[a,x2, ... , 251],[xm,1, ... , xzn]] is a law in G}. Let 1 = A0 <1 A1 <1 ... < At = G be the upper ¢n-series for G, where t is a positive integer. Since our characterisation of ¢;(G) holds 1 (Ia, g2, ... , gn],[gn‘_l, ... , SZn-U e A1 for all g2, ... , 32n in G] for 0 5 i 5 t-l. In particular, y2(yn(G)) = [yn(G), yn(G)] _C_ A for amr G, we may conclude 'that A+1 = {a e G: F t-l’ We claim in fact that y(1_1)m2(yn(G)) 2 A91 for l _<_ i 5 t. This has been shown for i = 1. Assume the statuent holds for some ~"-—— 1 5 i < t. Then by the induction hypothesis we have that Yint2(Yn(G)) = Y(i-l)nt2+n-l+l(yn(G)) 9:— At-(fl-l)’ H’m’ v(t_1)m_2(yn(6)) C; At.t = A0 = l and yn(G) is nilpotent of 61888 S (t-l)n+ lo Ihggran 2,22. For any G and n _>_ 1, G is dn-nilpotent if and only if dn_1(G) is nilpotent. 2322!- Since dn = [dn-l’ dn-IJ’ the sufficiency follows from Theorem 2.17. Hence assume G is dn-nilpotent of class t, where t is a positive integer. For each a e G define the words wo, wl, ... , wk, by "0 = wok) = a and HR = “1““) = wk“. 1&2. . 121.) = [wk_1, dk_1]. Let M g G, HC_Z G. By wk(H, H) we shall mean the subgroup generated by elments of the fem wk(h, m2, ... , mzk), whereheH, m1 cH for25i52k. Wenotewo(H, M) = H for amyM. We claim ‘13“) = {a c K: wn(a) is a law in K} for any group K. This is clear for n = 1, since dim) ‘-'-' 200 by Theorem 1.1+ and lit a e 200 if and only if w1(a, x) = [a, x] is a law in K. For arbitrary n we have d;(K)/Cx(dn_1(x)) =- dn:1(K/Cx(dn_1(l())) by Theorem 1.3(b). By the induction hypothesis, a e d;(K) if arr). only if wn_1(a) is a law mod CH(dn-1(K))' This is equivalent to saying wn(a) = [fin-1‘3» dn—l] is a law in I. Hence the claim is proved for aw group H. Let 1 = A0 4 ... 4 At = G be the upper dn-series for G. By the above claim, we know that A1 ={a e G: wn(a, G) Q A14} for l 5 i 5 t. Let H = dn_1(G). We claim that antlw) _C__ At-j for O 5 j 5 t. This is evident for j = 0, so assume it is true for some 0 5 j < t. We further assert that y Jmum) Q new. a) for At-j’ this latter conjecture holds for {F O 5 s 5n. Since "D‘At-j’ G) = s = 0 by assumption. Hence assume it is true for some 0 5 s < n. (a) = [va 8,101). H] 91.. mum. G). asap] = wed-lut- j’ G) by the induction assumption an! the fact that H E d8(G) for O 5 s 5 11-1. Hence the latter conjecture holds. In particular, Vii-nun“) 5;; "nut-j’ G) <;_ lbw”) by our characterisation of At-j' Hence the first conjecture holds. Conse- quently, yuflm) _C; 1,,t = A0 = 1 and H = dn_1(G) is nilpotent of class 5 tn. W 2.23. If G is a group such that dn(G) is nilpotent, 1 < H 4 G, then 1 < H n dn:1(G). 21:921. By Theoru 2.22, G is dml-nilpotent. The corollary now follows frat Theorem 2.11. m 2.21}. A group G is yl-hypercentral if and only if G = 1, since y;(G) = 1. For n > 1, G is yn-hypercentral if and only if G is hypercentral, since the marginal subgroups for such words are in the upper central series for G by Theorem Lb. 15 However , although a dZ-hypercentral group is dn-hypercentral for n > 2 by Theorem 2.15, dn-nilpotence for some n > 2 does not imply dZ-nilpotence. Let G = S“. The derived series for G is 1 4 Va 4 A,+ 4 G, where V“ is the four-group. Hence d3(G) = 1 so that -nilpotent of class one. Since G' = d1(G) = A4 is not 3 nilpotent, G is not dZ-nilpotent by Theorem 2.22. ._. e G d3(G) is d The following lemma is evidently well-known. m 3.35. Let ¢ = ¢(::) = x2. Then ¢*(G) = {a e Z(G): a2 = l] for any G. 2;gg_. Let H== {a c Z(G): a2 = 1}. Clearly H is a subgroup of'G contained in ¢‘(G). Let a c ¢*(G). Then (ax)2 = x? for all x c G. In particular, a2 = 12 = 1. Furthermore, for arm 1: c G, a‘l'xa = axa = axaaor'l = 2(2):":L = 1:. Hence a c 2(G) and H = ¢*(G). Ihggzau §‘_2_6_. Let ¢ = ¢(x) = x2. Suppose G is ¢onilpotent of class n. Then G is nilpotent of class 5 n and has exponent dividing 2". 1 2:93;. By Lama 2.25, the upper e¢-series for G is an ascending central series of length n. Hence G is nilpotent of class 5 n. Since the quotients in the upper ¢-series for G are elmentary 2-groups, G has exponent dividing 2". Thyran 2432. Assume G is ez-nilpotent of class n. Then (a) e2(G) is nilpotent of class < n. (b) there is a sequence 008 14G14 ... 4Gn= G of subgroups of G such that 51+1/G1 is nilpotent of class at most three. (c) G is solvable of length 5 2n. (d) if every elment of G has odd order, then G is nilpotent of class 5 3n. 16 M. Part (a) follows from Theoran 2.12. Let G0 = l 4 G1 4 ... 4 Gn = G be the upper ez-series for G. Since Gi/Gi-l’ l 5 i 5 n, is in the variety determined by e2, it is nilpotent of class at most three by Levi's Theorem and thus metabelian. Parts (b) and (c) now follow. Kappe [9: Sat: III, Teil 1+] has shown that a right Engel element of length two and odd order is in the third center. Under the hypotheses of part (d), e;(G/Gi) _C_ 23(6/61) for o 5 i 5 n-l. Hence G is yu-nilpotent and thus nilpotent. In Theorun 4.16 we shall show that e§(G) (______ 23(G) for all G. Consequently the hypotheses in part (d) may be diapensed with ani we may conclude that arw ez-nilpotent group of class n is nilpotent of class 5 311. W _2_,2§. Define H¢(G) = H¢ = {a e G: «81, ... , gn)'l¢(g1, ... , agi, ... , gn) c H for every 1 and g1, , gninG} tobetheWoinnG. It is not difficult to see that H¢(G) C; G a!!! that H 4 G implies that H L; H¢(G). Ihgqran 532. If G is ¢-hypercentra1 and H < G, then H¢ g H. In particular, if also H 4 G, then H < 11¢. m. Let G0 = 1 4 G1 4 ... 4 Ga = G be the upper ¢-series for G. There is an ordinal e such that GB Q H, G8+1 (Z; H. Choose a e Gan, a é H. We know that GB'tl/GB = ¢'(G/GB). In particular, for g1, ... , gn in G, we have ¢(g1, ... , gn)'1¢(g1, ... , agi, ... , gn) is in GB g, H, since a c Gad-1° Hence a c H¢(G), a t H. m w. Perhaps a more natural way to define a ¢—normalizer H(¢, G) of H g; G is the set a c G such that 17 ¢(h1, ... , ahi, ... , hn) c H for every 1 ard hl, ... , hh e H. This ¢-normaliser is NG(H) for ¢ = y2, but it is not necessarily a subgroup for every word ¢. From the definitions we have that H¢(G) is a subset of H(¢, G). Hence by Theorem 2.29 we may conclude that a proper subgroup of a ¢-hypercentral group is a proper subset of H(¢, G). 3223;}; 5.11. For any word ¢ we may define a BM group G to be a group possessing a finite normal series A0 = l 4 A1 4 ... 4 an = G such that Aid/A1 = ¢"(a1+l/ai). It is clear that a ¢.ni1potent group is also ¢esolvable. We may equivalently define a group G to be ¢-solvab1e if and only if the descending series G D ¢(G) D ... D ¢n(G) 3"“ D WWI-(G) == ¢(¢n(G)) D ... reaches 1 in finitely many steps. A group is yzesolvable if and only if it is solvable. Many of the standard theorems for solvable groups can be generalized to ¢-solvab1e groupS. CHAPTE III MARGINAL AUTOMORPHISMS In this chapter we will consider two generalizations of the _a u marginal subgroup of G--one in G and the other in Aut(G). r Definition 1;. Let g) = ¢(x1, , xn) be a word in n variables, G a group. Define ¢°(G) = {a c G: ¢(gl, ... , 3;, ... , gn) = ¢(g1, ... , g1, ... , 3n) for all choices of g1, ... , gn inG and i=1, 2, ... ,n} tobetheWMochorrespording tc¢. 2m 2&- In any group G. (a) ¢°(G) e Char(G). (b) if K/¢"'(G) = Z(G/¢*(G)), then Z(G)K ; ¢¢(G). In partisan. Ms) E; ¢°(G). (c) [¢°(G). ¢(G)] = 1- (d) ¢°(G) = {a cG: ¢(g1, , agi, , gn) = ¢(g1, ... , gia, ... , gn) for all choices of g1, ... , gn inG and 1: 1, 2, , n}. Erggf. (a) Let a, b c ¢°(G), g1, ... , gn in G. Then 1b ¢(819 0-0 9 8;. 9 H- 9 Sn) = ¢(819 0-0 9 8:41, 0" 3 8n) 3 ¢<81s 0-0 s (8;-1)a9 0-0 a 8n) = ¢(81, eee , 81, cos , 8“) for each 1. Hence a'lb c ¢°(G) and ¢c(G) E G. ac Now let 0. e Aut(G). Then ¢(g§, ... , (3g) 8%) = ¢ 9. make» 1:. mm». In partisans. ¢ 1, ¢ = [1, 6], where w().), w(9) are less than w(¢). By the induction hypothesis, {(6) = A'(6), 3(6) = 0'(G) for all G. Consequently, 8(G) = ¢'(6) for all G by Theorem 3.15. Mp}; 1,12. The conclusion of Corollary 3.16 does not r hold for ¢ = ¢(x) = x2. By Lanna 2.25, ¢*(G) = {a c 2(6): a2 = 1] for em G. Let 6 = S The set H of squares of slanents of G is the 3. derived group of G, the normal subgroup of order three. Since 6 is centerless, we may consider 6 to be a group of automorphisms on itself. Certainly H as a subgroup of Aut(G) fixes each element of H. Hence 1 1 HQ 8(G). But ¢'(G) = {a e Aut(G): [G, a] Q ¢"'(G) = 1} = 1. We recall that a c yé(6) = QC) if and only if a is a normal automorphism; that is , a comutes with the inner automorphisms. Franklin Haimo [3] and W. E. Deskins [l] have considered higher mrmal automorphisms. In particular, define T1(G) = 6(6) and Tn(6) = {a c Aut(G): xa .- 3: mod Zn(6) for all x e G} for n > 1 to be the my; W of 6. By Corollary 3.16 we may now add that Tn(6) = yn;1(G) = Inn(G) for n 2 l. madam 2.11;: Lot 9) = we = x“. n .>. 1. Then ‘v'zm n 3(a) = ;2(6) n ¢'(6). If n = 2, then‘also ¢'(6) _C; 'y-z(6) so that ?2(G) n We) = ¢'(6). £3991. Let a e ;2(6) 0 3(6). Then (xn)‘ = x1'1 for all x e 6. This implies l = (xn)"]'(xn)a = Ex", a] = [x, a]n, since [x, a] c 2(6). Hence a c ¢'(G). By Theorem 3.9(a), we have 72m) 0 3(a) = ¢e(o) 0 372(6). If n = 2, ¢'(G) = {x e 2(6): 2:2 = l} . Hence ¢'(G) Q ?2(G). 25 W 3, 2. Assume 6 = S + T, T is torsionfree Abelian and S = I: {: c. c I}, where o(aa) is a prime power. Let ¢ = ¢(x) = x". Asstme further that if ad has order pzu, then ma 2, 239 “11°” P; is the pa-share of n. Then ¢*(6) = 2 {: a c I} and ¢'(G) = 5(6) is Abelian. m. Since G is Abelian, ¢'(G) = 6(G) by Theorem 3.18. Also, .—-—.- ¢'(G) = ¢"(T) + 2{¢'()z c c I} = >:{¢"'(: xn = l} = {x c : xp 1} = (a >. Write n = pst, where (p, t) = 1. Since also (o(a), t) = 1, there is a s s b c such that a = bt. Hence ap = bt'p b11 and ¢"'() = <(aPs)Pm'28> = <(me'28)“>. Then ¢'(6) stabilizes the series 1 4 ¢'(G) 4 6. Consequently ¢'(G) is Abelian. ' my 1.39. Let G = ”“196 c c 1), ¢ = ¢(x) = x2. By Corollary 3.19, ¢'(G) = 6(G) is Abelian. Since f c Aut(G) defined by x1‘ = x'1 is in ¢'(G), we have that ¢'(G) 7‘ 1. We also note that if o(I) = 1, then ¢'(6) = 3(G) = Aut(G) ’-l-’ J2. Also, for ¢ = x2, if 6 is torsionfree Abelian or periodic where each elanent has odd order, then ¢'(G) = 3(6) = l by Corollary 3.19. Furthermore, if 6 is Cornikov complete, it is generated by its n-th powers so that ¢'(G) = 3(6) = 1 for ¢ = ¢(x) = x". CHAPTER IV THE HEEL mm In this chapter we shall investigate the marginal subgroup for the Ehgel word e2 of length two. We note that Theorem 1.3(b) does not apply, since e2 is a commutator with a repeated variable. The metabelian margin d;(6) will play a role with e§(6) for each 6, so we will also derive some results for d2 and, where possible, «Item than to am solvable word dn' By "Bagel word" we will mean ”Rigel word of length two“. For any a we will write 14 a M(6) a dam.) and E - 3(a) = e;(6) for the metabelian and mel margins of G respectively. I‘m in.- In W group G. (a) d;(G)/c5) = d;1(G/ca(dn.1(o))). In particular. 14(6) = {a c 6: [[a, x], [y, 3]] is a law in 6}. (b) awn/2(a) g; d;(6). In particular, 23 _C_ u. (c) [11, dn(6)] g; Z(dn(6)) for nz 1 and (M, G] ; 2(6'). m. (a) This follows from Theorem: 1.3(b), since <11,1 is an outer mutator word for each n. We note in particular that M/CG(6') = 2(6/CG(6')). The second statement was verified in the proof of Corollary 2.22. (b) We irduct on n. For n -= 1, 21(6) E dam) = 2(6). For n > 1, let 3 s 6/Co(dn_1(6)). Then 3:365 . dn:1(6) _D_ 26 2? zn(n_1)/2(E) by part (a) and the itduction hypothesis. Mthermore, [zn(nf1)/2(G)’ n(n~l)/2(6)] ; zn(nO-l)/2 - n(n-l)/2(G) = Zn“) ”‘1 [2156). dmlflifl Q [Zn(G). vn(G)] = 1 so that [zn(n|'1)/2(G)’ “(151),2‘Gn Q CG(dn-1(G))° CommmtJ-Ya anrfi-lflzm _C'__"_ zn(n—l)/2. M. Dew. Then there is an m c M such that it” 3' x for all x e dn(6), x 7‘ 1. Define fm as in Lanna 4.2. For x, y e dn(6), fm(x) = fm(y) implies that a4»? = y‘lf' or that yr’l = (37:61)”. By assmnption, x = y and 1’m is an isomorphism from dn(6) onto a subgroup of Z(dn(6)). Hence dn(6) = 2(dn(6)) and dm1(6) = l, a contradiction. We now turn to the Ehgel margin E. For convenience we define El=(a c6: [6, y, y]=[x, y, y] forallx, yeG) ardL(6)= {a cos [a, x, x]= 1 forallx cc} to bethe subgroup of right Rigel elasents of length two. It is not difficult to show that E 5; El c Char(G). We will need the following lama. For the proofs of the various parts, see [9]. 29 m M. (Kappe) In amr group 6, where a c L(6), g, h e 6, (a) L(G) c Char(G). (b) [a, g, h] = [a, h, gj'l. (c) [a, [3. 11D" [as so b.12- (d) [a, g, [h, g]] = l. (e) if a has odd order, then a c 23(6). mgr-,1. Every 3 c E(G) is both a right all a left @301 el-aent. In particular, 22(6) _C_Z_ 133(6) ,9; L(6). m. Let x c 6. Then [x, a, a] = [x, l, l] = l and [a, x, x] = [1, x, x] = 1. Hence a is both a left and a right Engel element of length two. Furthermore, e2(6) g; y3(6) for all 6 implies that y3(G) Q e203) for all 6 by Theorem 1.5. By Theorem 1A, y;(6) = 22(6) an! the result follm. M2 is. In any group G. (a) £1 = {at [a, x] c 060:6) for all x c G} = L(6). (b) [a, x] c 660:6) n CG(a) for all a a El, 1: c 6. Furthermore, [a, 12]" = [a,, 1:8] for all integers r and s. (c) aG and xEl are Abelian for a c £1, at e 6. (d) E31 _C__ I, where I = fl{CG((xG)'): x e 6} <1 6. m. (a) Let a c 31. Then [ay, 3:, x] = [y, x, x] for all x, y c 6. This is equivalent to saying that l = [[ay,ny, le, x] = [[a, x]y[y, ny, xJ'l, x] = [[a, fly, x] for all x, y e G. Since at and y are irdependent, we may conclude that a c El if and only if 1 = [a, x, xy] for all x, y c 6 or, equivalently, [a, x] e 660:6) for all 3:. That 31 _C; L(G) follows from [a, x, at?) = 1 by letting y = 1. Conversely, let a e L(6). We have, for x, y c 6, [a, x, 15'] = 30 [a, x, xfx, y]] = [a, x, [x, y]][a, x, x][x’ y]. By the definition of L(G) we see that [a, x, x] = 1. By Lemma 4.6(d) we also have that [a, x, [x, y]] = 1. Hence [a, 1:, xy] = l and a e E1. (b) Since a is a right angel element, we have [a, x] e CG(a) by [9: Lemma 2.1]. Part (a) says that [a, x] c cG(xG) for all x c G. The remainier of part (b) follows from [18: Theorem 3.1%]. (c) From part (b) we see that a" = a[a, x] e CG(a), since Ea a and [a, x] are in CG(a). This implies that aG is Abelian. ' The proof that xEl is Abelian follows similarly from x‘1 = xfx, a], [x, a] c CG(xG) Q CG(x). (a) By part (b) we may conclude that [a, xy] c CG((xy)G) = L" °G("G) for all a a El, x, y e G. We claim now that (xG)' = s, where s = < [36“, x']: w, z c G >. Clearly s _C_:_ (xG)'. A generator a of (xG)' may be written a = [(xflm (xfl'Wm, (36”)le (filflnj for elements y1 c G. By [18: Theorem 3.4.2] we may write a = 11 [(xflfli, (xfl)y3]"i,3 for 2.1.3 a 1:6. .But 2:9 <1 G, so we may assume (xG)' is generated by elements of the form [(actl)“, (film for various w, s a G. Then [(x'l)", x‘] = ([x", x‘]'1)(x'1)w = ([x“, x"])(x‘1)" is in 5, since 8 o G. Similarly, we may show that [x", (x'1)”] and [(x'l)", (x'l)z] are in 5. Hence the claim follows. Let a e E1. By Lemma n.6(c), we have [a, [x", x‘j] = [[a, x"], x212 = 1. By the claim this implies that a c CG((xG)'). Furthermore, xG <1 G, (xG)' o Char(xG) for each x imply (xG)' <1 G. By the u/c Theorem, CG((xG)') <1 G for each x c G. Therefore I <1 G. 31 , Thgrem 542. In any group 6, E = E(G) = (a c G: l = [x, a, y][x, y, a] for all x, y c 6}. m. Set 1?.2 = {a c 6: Ex, ay, ay] = [x, y, y] for all x, y c 6}. We know El = (a c 6: [a, x] e CG(xG) for all x c 6} by Theoran 4.8(a) and E = El 0 E2. Let S be the set given on the right in the statanent of the theorem. Suppose a c S, x e 6. Then 1 = [x, a, XIX, x, a] = [x, a, x]. This implies that a c El = L(6). Since also E E E1, it suffices to show that E 0 E1 = E1 0E2 ’- E108. Then, forx,yc6,acElnEzifand onlyif EX. 1!. fl = [X. ay. ay] ' [1. W. 111:. ay. at]y = [[x, y'l[x, 3])” VIE"! 5’1"» fly: fly x, a y x, a y E 3m. aJ’. ylx. r. alt “as. ally. sly. By assumption, [a, x] c CG(xG). Since CG(xG) <1 6 by the N/C = [xi Y9 Y] Theorem, we also have that [a, fly a CG(xG). Consequently, conjugation by [x, a-Iy is irrelevant in the last statment above because all the commutators are in 1:6. Therefore the above is equivalent to [x, Y! fl = [x, 3'9 VIE", sly: YIX’ 3" a.1311"! fly, 31y 01' l = [x, a, y][x, y, a][[x, fly, a] for all x, y c 6, a c E(G). Now a arxi Ex, ajy are elanents of 9.6. By Theoran 4.8(c), aG is Abelian. This implies that [[x, fly, a] = 1. Therefore E(G) is contained in the set S. We have already shown that S is a subset of E1 = L(6). Consequently, all the above argments are reversible and we may conclude that s = E(G). 32 9.933453: 3.1.9- In are group G. E(G)flCG(G') = 22(6). Proof. We need only verify that E(G) n CG(G') (_:_ 22(6) by Leanna u.7. Let a c E(G) ncG(G'). By Theorem n.9, 1 = [x, a, y1[x, y, a for all ,x, y o G. But a c CG(6') implies that [x, y, a] = 1 and thus that [x, a, fl = 1 for all x, y c G. Hence a s 22(6). W 353- (a) Suppose a c L(6), x, y e G. Then [)9 a. y] = [aa 3'. X.)- (b) E(G) = {a c G: [x, y, a][a, y, x] = l for all x, y c G). zgo_o_f. (a) [a, y, x] = [a, x, y]'1 by Lemma 4.6(b), = [be «0-1. yTl 7;. "f\ v. 0 [as x] = «[x. a. y1'1r1) = [x, a, y], since [a, x] c CG(xG) by Theoran 4.8(a). (b) Let S be the set given on the right in the statanent of part (b). By part (a) ard Theorun 4.9, we have that E(G) is a subset of 3. To prove the opposite inclusion, we need only show that S is a subset of L(G) and then use part (a). Suppose a c S, x e 6. Then [x, x, a'][a, x, x] = [a, x, x] = 1. Hence a c L(G) and the theoren follow. M £42. In arw group 6, [14, 6] 013(6) 2 22(6). 23:393. By Lanna 4.1(a), we have M/CG(6') = 2(6/CG(G')). Hence [M, G] n E E CG(6') n E = 22(6) by Corollary Lalo. 1.12222; 5.13. Let a e E(G). Then [a, 6, G13 = [a3, G, G] = 1. 33 m. Let x, y c 6. By Theoran 4.ll(b), [1, y, 31;, y, x] = 1. Then [x, y, a] = [a, [1, y']]'1 = ([a, x, ysz)":L by Lama 4.6(c), = [a, y, x]2 by Lena b.6(b). Hence 1 = [a, y, {[a, y, x] = [... y, xfia, y, x] = [a, y, x13. By Theorem 4.8(c) we have that a6 is Abelian. Hence [a, x, y]3 = 1 for all x, y c 6 implies [a, 6, 6] has eXponent dividing three, and [8, x, y]3 = [‘3’ X, y] B 1. Pa"; The followirg two corollaries are immediate from Theoran “.138 922% 4,1“. For any group 6, 13/22 has exponent three. W 1+, 5. If E has no elasents of order three, then E = 22. : m m. In any group 6, E(G) _C_ 23(6) _C; 14(6). In addition, E/Z2 is an elmentary Abelian 3-group. 2m. We need a slightly stronger result than our Lama n.6(e). In his proof, Kappe [9: Sate III, Teil b] shows that for a c L(6), x, y, s c 6, [a, x, y, 15]“ = 1. Since aG is Abelian by Theoran n.8(c), we may assert that Ea“, x, y, s] = l or a“ s 23(6). By Corollary ml», for a c E _C; E1 = L(6), we also have a3 e 22(6) _C_ 23(6). Hence a = a“(a3)-1 c 23(6) and E(G) 9: 23(6). It follows now from Lemma n.1(b) that 23(6) g M(6). The second statement then follows from Corollary 4.1“. 99mm £5.12. In am group 6, E(G) is nilpotent of class no greater than three and metabelian, and [a, E, B] has exponent three. Nrthemore, if CG(6') _C_ E(6), then M(6) = 23(6). M. The first statement follows imediately from Levi's Theorem, since e2 is a law in E(G). Alternatively, we may conclude that E(G) has nilpotence class no greater than three an! is metabelian 31+ from Theorem: 4.16. By the same theormn, we know that [E, E, E] g; Z(E) is Abelian. Hence Theoren 4.13 implies that [E, E, E] has exponent three. Suppose CG(6') _C_; E. By Corollary 1+.lO this implies that CG(6') = 22(6). From Lemma n.1(a), M/CG(6') = 2(6/CG(6')). Hence 14(6) = 23(6). heorem 3,18. (a) [6', M, E1] = [6', E1, 11]=[1~1, G, 6'] = 1. (b) [G, 14', E1] = 04', E1, G] = [G, G, M'] = 1. In particular, [M', E1] _C_ um. 2:22;. (a) By Lemma 4.1(c), [11, G]g2(G') so that l = [11, G, 6']. Now let a e E1, m c 11, x c 6'. By Lemna n.6(c), [a, [m, x]] = [a, m, x]2 = 1. This implies [6', M, E1] '-" 1. By the Three Subgroups Lamas, we also have that [6', E1, M] = l. (b) As in the proof of part (a), M' _C_:_ 2(6') so that l = [6, 6, M']. Let a e E1, x e M', g c 6. We have [a, [g, x]] = [a, g, x]2 = 1. Hence DU, 6, E1] = 1 ani, as above, [M', E1, 6] = 1. W 3.12. Suppose M'/F(A) is divisible, where A g; Aut(M'), A g ElCG(M')/CG(M') and F(A) = {x c M': xa = x for all a c A} is the set of points fixed by A. Then (a) [M', a] is divisible for each a e A. In particular, [111, E1] = [M', a] is divisible. (b) for all e e E1, 3: c M', f°(x) = [x, e] defines a homo- morphism from M' onto a direct divisible sumard of 2(6). (c) for all e e E1, m c M, there is a homomorphism t = t(m, e) from 6' into a direct divisible summam of 2(6) such that {x 126': [x, m] cM'}g_Ker t. 35 22221:. By Theorem 4.18(b), [111, El] _C; 2(6). For e c F1, let a c A be the automorphism on 111 corresponding to eCG(M'). Then M'/F(a) ’é’ [111, a] = [111, e] _C; 2(6), where F(a) = Ker f. and [111, a] = f°(M'). Since M'/F(a) ’-‘-' (M'/F(A))/(F(a)/F(A)) is divisible, we have that [111, e] is a divisible subgroup of 2(6) and hence a smumand of 2(6). Barthermore, [111, El] = < [111, e]: e e E1 > is divisible because it is generated by divisible subgroups of 2(6). Parts (11) and (b) now follow. Let a e E1, 111 c M. Define the homomorphism fo from 11' onto [M', e] as in part (b). We know 11' _C_‘_ 2(6') and [M', e] is divisible. Since then [M', e] is an injective Z-module, fe may be extended to f: c Han(Z(G'), [111, e]). Define f1m c Hom(G1, Z(6')) as in Lema 4.2(a). Then t = f';fIII c hom(G1, [111,e]). If x c 6' such that [x, m] e 111, then t(x) = fzfmu) = f;([::, m]) = f°([x, m]) = [x, m, e] c [6', M, E1] = l by Theoren 4.l8(a). mg 4,20. By Theoran 4.16, E _C_ Z so that [65 E] _C; 2(6). 3 Then for all a c E, x e 6', fa(x) = [x, a] defines a homomorphism from G1 into 2(6). We shall investigate the action of E ani E1 on 6'. By Runark 4.4 we know that M/CG(6') acts as a group of Abelian central auto- morphisms on G1. Then (1;1 rho/(1‘10 CG(6')) C; M/CG(G') is also such a group. Let A2 Q Aut(G') be the corresponding group of auto- morphisms. Furthermore, E/Zz = (E n M)/(E n CG(G1))(; A2 by Corollary 4.10 and Theoran 4.16. Let A1 _C_ A2 be the corresponding group of automorphisms on 6 ' . 36 Ihggren 442;. (a) If Encp(Z(6')) = n < 19, then Eltp(A2) | n. A (b) If G1 is a p-group, A Q A2 is periodic, then A is a p-grcup. (c) If G1 is polycyclic, then A1 ’5 13/22 is finite. m. (a) Suppose 2(6') has exponent n. Then, for x e 6', c c A2, 1 = [x, 0]" = [x, 11“] by Theorem 4.8(b). Consequently, c“ = 1 an! A2 has exponent dividing n. (b) New assimie A is periodic. By Theorem 4.18(a) we may conclude that [G1, 11, El] -- [G1, A, A] = 1. Thus A stabilizes the nomal series 1 c [G1, A] <1 6' of 6'. By the arguments used in [28 Corollary 5.33], we have that A is a p-grcup. (c) Smirnov [19] has shown that a solvable group of automor- phisms of a polycyclic group is polycyclic. By Thecru 4.16 we have that Al is a finitely generated periodic Abelian group. Hence A1 is finite. m $33. If Z(G)n 2(6') has no elements of order three, then E = 22. m. We shall show that A1 = 1. Let a c A1. By Theorem 11.16, E g 23. Hence [G1, E] = [G1, Al] ; zccm 2(G1). Them, by Corollary 4.14, 1 =- [x, c3] = [x, c]3. By hypothesis, this implies that 1 =1 [x, (1]. Consequently c. = l. m .4423. Suppose A2 3‘ l is not torsionfree. Then 6' has a proper subgroup of finite iniex. m. For 1 =f a c A2, the homomorphism from 6' into 2(6') defined by fa(x) = [x, a] for each x c 6' is nontrivial. We may fini an a e E10 M \ Elf) CG(6') such that (x, a] = Ex, a] for all x 116'. If a has finite order, then there is an integer n such that an e CG(6'). 37 Thus 1 = [x, a]n = [a:, an] and 6'/I(er f0L “=’ fa(6') C; z(G1) is a non- trivial direct sum of cyclic groups each of order bourded by n. In particular, there are subgroups H and C of 6' such that 6' IKer fa = H/Ker fa + C/Ker fa and C/Ker fa is nontrivial and finite. Consequently H < G1 and 6'/H é C/Ker fa is finite. M ‘_+_,_2_4. If E > 22, then 6' has a proper subgroup of finite iniex. f— 2.119.111 If E > 22, then A1 is a nontrivial torsion subgroup of A2 by Theorem 4.16. Hence A2 # l is not torsionfree and the theorem 3 appli e8 . It is known that no complete, or even Cernikov complete, group L can have a proper subgroup of finite irrlex (see [10: p. 234]). The following two corollaries come directly from this fa ct. Cgrolgn 4,25. If 6' is Cernikov complete, then E = 2.2. gm 4,26. If 6 is metabelian ard 6' is divisible, then E = 22. m 4,32. If 6'/(6' fl CG(Eln M)) is periodic, then A2 has no clanents of infinite height. m. Suppose a 11: El HM \ E1 066(6') is such that the irduced automorphism a c A2 has infinite height. Let x c 6', and seems there is an integer n such that xn c CG(E1 H M). By hypothesis, there is a B c A2 such that c = a“. Thus we may find b c 131 mm Elf] CG(6'), y e E1 0 06(6') such that a = bny. Then [x, a] = [x, buy] = [x, b”] = Elan, b] = l by Theorun 4.8(b) and a 1: El 0 66(6'), contrary to assumption. We note that since A1 has bounded exponent, it can have no elements of infinite height; certainly no nontrivial element of A1 38 can be divided by 3 = mpul). Similarly we may show that A2 has no elqnents of infinite height if 2(6') has bounded exponent by Theorun h.21(a). Theorgg M. Assume 2(6') is torsionfree. Then A2 is torsionfree. firm. Let 1 3‘ a e A2, o(a) = n < 0°. Then there is an x c 6' such that 1 4 (x, a] c 2(6'). But 3, c1“ = 1):, an] = 1 so that o([x, a1) I n. Hence a has infinite order. Since A1 Q A2 is torsion, we must have that A1 = 1 or E = 22. The proof of Theoran 15.29 is an improvement on a result of Turner-Smith [22: Lanma 3.2], who has shown that for an automorphism group A on 6, A has to have bouxded exponent whenever the set {[x, c‘l: x c 6, a c A} is finite. Theoran ‘_+_,_2_2. If a e E1 is such that Sa ={[x, a]: x c 6') is finite, then aZ(6) c ELI/2(6) has finite order. The orders of all such elements are bounded if the cardinalities of the sets are bounded. £3933. Assume that Sat has n elements, 3: c 6'. Then at least two of the elements [x, a‘], [1:2, a], ... , [xml, a] are the same. Hence there is an integer k = k(x) _<_ n depending on x such that [x, ak] = [xk, a] = 1 by Theorem n.8(h). Since the k's are bourded by n, we may assert that [x, an!) = l for all 3:. Consequently an: e 2(6). M {51. We have proved that if 2(6) 0 2(6') has no elments of order three, or if 6' has no proper subgroup of finite irdex, then E(G) = 22(6). We shall now show that there exists a group 6 such that 22(6) < E(G) < 23(6). 39 Let H = < a1, a2, a3: 2:3 >. Levi and van der Waerden [11] have shown that H has nilpotence class exactly three and satisfies the law e 2 class at least three having no elments of order three (see for example . Hence E(H) = H = 230-!) > 2201). Let K be amr group of nilpotence [17: p. 198]). By Theorem n.22, E0!) = 220:) < Z300 _C; K. Let G = H x K. By Theorem 1.2, we have E(G) = £201) 3: E(K) = H x 2200. Consequently 22(G) < E(G) < 23(G). We have also shown that E E; El = L(G). Define NA(G) = fl {NG(H): H maximal Abelian subgroup of G} to be the A-Norm (or Abelian-Norm) of G. Kappe [9] introduces this concept and proves that a c NA(G) if arr! only if [g, h] = l for g, h c G implies that [a, g, h] = 1. By Corollary l&.10 it follows immediately that E g NA(G) 9; E1. Definition m. We shall. say that a word ¢ satisfies the W m if [G: ¢"(G)] = m finite implies ¢(G) finite with order which divides a power of m for all groups G. Schur showed that Y2 satisfies the Schur-Baer property; Baer eacterded this result to am outer commutator word 53 (see [20]). We shall need the followirg theoran. For a proof (due to P. 31:11), see [20: Theoran 2]. Iheorun 343;. If ¢ generates a locally residually finite variety, then ¢ satisfies the Schur-Baer property. M £51}. If ¢ c fez, e3}, then ¢ satisfies the Schur- Baer property. m. Suppose ¢ = e2. A group in the variety generated by ¢ is nilpotent by Levi's Theoran. A finitely generated nilpotent group 40 is residually finite by P. Hall [6]. Consequently a finitely generated group in the variety generated by ¢ is residually finite and Theoran “.33 applies. Let ¢ = e3. Heineken [8] has shown that a group in the variety generated by ¢ is locally nilpotent. Hence a finitely generated group in this variety is also residually finite ard the theorun follows as above. £922.33 94%. We note that in P. Hall's proof of Theorun ‘0.33 we may sharpen the result somewhat if we put some restrictions on G itself. That is, if ¢‘(G) is locally residually finite for all G in some class 2 such that 2 = (Q, 8):, then ¢ satisfies the Schur-Baer property for all G c 2. In particular, we have the following: Theoran 9415. If G satisfies the maximum or the mirdmm coalition, or if G is an SN"l group, then en satisfies the Schur-Baer property for G. 23:22:. Suppose G satisfies the maximum condition. Then, by [17: Theorem V1.8.J], we have that the set of left Ehxgel elements (of all lengths) is the Hirsch-Plotkin radical R. Since then e;(G) _C_ R is locally nilpotent, it is locally residually finite. By Ranark “.3“, en satisfies the Schur-Baer property for G. Vilyacer [23] has shown that an angel group satisfying the minimum condition is locally nilpotent. Plotkin [115] has proved that an mgel group which is also an SN"I group is locally nilpotent. Hence the ranainier of the theorun follows as above. P. Hall has made the following three conjectures concerning arbitrary words ¢ ani groups G (see Turner-Smith [21]): ul I. If ¢ is finite-valued in G, then ¢(G) is finite. II. The word ¢ satisfies the Schur-Baer property. III. If G has the maximum condition on subgroups and ¢(G) is finite, then G/¢"(G) is firdte. It is not known whether these conjectures are universally true. We have shown that Conjecture II is satisfied for ¢ c {e2, e3}. Our results are more limited for these words ani the other two F— conjectures. We shall need the following lama. The arguments follow those .'M A EL'11“_*. used in [22: Praposition 1]. E M M. Suppose G is in a class of groups in which Conjecture II is satisfied locally for ¢. If G is also locally residually finite, then ¢ an! G satisfy Conjecture I. 21:29.: Assme ¢ is finite-valued on G. Then there is a finitely generated subgroup H of G such that ME!) = ¢(G). Since ¢ is finite-valued on H, we have that the set of eluents of the form t =- ¢(h1, , ahi, , hn)'1¢(h1, , hi, , hn) for a, h1 c a, 1 g i 5 n, is finite. Lot 121, see , tk be the nontrivial values. By hypothesis, H is residually finite. Hence we may find I. <1 B such that H/L is finiteaniti #1., 15151:. Leta cL, 111, , hncH. Then w= ¢(h1, , ahi, , hn)'1¢(h1, , hn) eL. Since no such nontrivial elment can be in L, we must have that w = l and L _C; ¢‘(H). Hence H/¢"(H) is finite ani, since Conjecture 11 holds in H, we have that ¢(H) = ¢(G) is finite. m m. If ¢ c {e2, e3} arrl G is locally residualJy finite, then ¢ and G satisfy Conjecture I. #2 25923:. By Theoran “.33 we know that ¢ satisfies the Schur- Baer property. Hence Conjecture II is satisfied for ¢ in amr group ani the result follows from the lunma. 3m}; 9.1.3.8; Conjectures I and III seem quite difficult to verify for Engel words. Conjecture I has been substantiated for nilpotent ard solvable words but not for outer commutator words in general (see [16] and [21]). Conjecture III was proved for any outer commutator word by P. Hall (see [21]). We note that Theorem “.36 verifies Mersljakov's [12] variant of the last conjecture for finitely generated residually finite groups G and ¢ c {e2, e3}: IV. If ¢ is finite-valued on G (where G does not necessarily satisfy the maximum condition), then G/¢“(G) is finite. Merzljakov [12] proves that for an arbitrary word and an arbitrary linear group over a field all four conjectures are true. We note also that Turner-Smith [22] has shown that all three conjectures hold for every word ¢ and every group G in the class of groups whose homomorphic images are all residually finite-mfor example, the class of polycyclic groups . Iheorg £532. If G satisfies the maximum condition ard yn(G) is finite for some n, then G/en:l(G) is finite. 22292.: By Remark 4.38 we know that Conjecture III holds for yn. Hence G/y;(G) is finite. That y;(G) _C_:_ °n:l(G) follows from the fact that en_1(G) _<‘;_ yn(G) ani Theorem 1.5. Hence Glen:1(G) is finite. BIBLImRAPHY 9. 10. 13. 11+. BIBLImRAPHY Deskins, W. E. "Automorphisms, multipliers, and factors." Report to the Ohio StatenDennison Conference, in March 1969. Gorenstein, Daniel. Einitg Grougg. New York: Harper and Row, 1968. Haimo, Franklin. "Normal automorphisms and their fixed points." (Unpublished . ) Hall, P. "Verbal ani marginal subgroups." W math“ 182 (19%), 130-141. Hall, P. "Nilpotent groups.” Report to the Canadian Kathe- matical Congress, 1957. Hall, P. "On the finiteness of certain soluble groups." 2:33, W (3). 9 (1959). 595-622. Hall, P. and B. Hartley. "The stability group of a series of subgroups." Proc, Londgn Math, SgcI (3), 16 (1966), 1-39. Heineken, H. "Ehigelsche Elmente der Lang. 41-01." Hilflfls We 5 (1961), 681-707, Kappe, Wolfgarg. "Die A-Norm einer Gruppe." 5 th , 5 (1961) . 137-197. Kurosh,6A. G. Ihe Ihggm of. Groups, Vol. II. New York: Chelsea, 19 0. Levi, Friedrich and B. L. van der Waerden. "fiber eine hesondere Klasse von Gruppen." bhand t Sem , Mersljakov, Ju. I. "Verbal and marginal subgroups of linear groups." Soviet Math, ng1,, 8 (1967). 1538-15‘t1. Neumann, Hanna. Varieties Grou . Ergebnisse der Mathmatik uni ihrer Grenzgebiete N. o. , Band 3?. New York: Springer-Verlag, 1967. Plotkin, B. I. "On some criteria of locally nilpotent groups." 5 am at 8131‘ (Ne Se), 9 (195“), me 3’ 181-1860 (Russian. “3 15. 16. 17. 18. 19. 20. 21. 22. 23. ha Rhantulla, A. H. Unpublished lecture notes. Rosenlicht, M. "On a result of Baer.” Prgc, Amer, Math, Soc“ 13 (1962), 99-101. Schenlcnan, Eugene. Gr u Theo . Princeton, New Jersey: D. Van Nostrarxi Comparw, 1965. Scott, W. R. Gr u T r . Ehglewood Cliffs, New Jersey: Prentice-Hall, l . Smirnov, D. M. "On groups of automorphisms of solvable. groups." m: 32 (7“) (1953). 365-38“. (Russian.) Stroud, P. W. "On a property of verbal and marginal subgroups." EEC. Cammgge Phil, 39c“ 61 (1965), 141-98. Turner-Smith, R. F. "Marginal subgroup properties for outer commutator words." Eroc, London Hath, Soc, (3), 11+ (1964), 321-341- lI‘urner-n‘iamith, R. F. "Finiteness corditions for verbal subgroups." Journal Lgflgn flath. 59c“ 41 (1966), 166-176. Vilyacer, V. G. "On the theory of locally nilpotent groups." Us hi Ma Na (N. 8.), 13 (1958), no. 2, 284-285. (Russian.