.- u . .. ‘3 7H . . . . 2.. ..._....:.:. . . .1... ._..... .51.: ..... .. . . ...,........ .......... ......... Ty. . ...: .a................~.........‘.... 2... ... ..... .. ...:.... .. .... ....u........ :3 3. 7......3... ... ................3 In... 3..... .T .1....--.................. .3....:..........:.......3.3. z... ....... ..... .r... . .31... .1... y. x. 4.....AA..~‘|... ._..:. . . .. . aw...“ 4 .. 05‘ s Anew 32950555 5 -,~ 1.1;. x .. . 5.. 5. . i .1531..- 3.3...-............,...-..u.....3. ., .._......5 5.1.3.3... ..........:...... .58 v I— lv“..-! I‘afiitldId'A. {'53 ‘13.!) r .5? ;.rrrv\tr.l .1. q .1... . 1. .... . 5 . ..-. . 2. . . . 5.: . 1 :3:=.:._,—..3=- 3.5.5. ... I»... LIBRARY ‘Ildiingc-O U' . This is to certify that the thesis entitled SPECIALIZATIONS OF RESIDUAL CENTRALITY 'IN GROUPS presented by Roger Dale Konyndyk has been accepted towards fulfillment of the requirements for Ph. D . Mgr”. in Mathematics ‘ V \ 1 ,/ ,1 5 ', , , 7 ,1 . It ,/ ,7 // E w I Jr'v/r’y.’ Major professor ‘ ' 0-7639 1% ABSTRACT \"SPECIALIZATIONS OF RESIDUAL CENTRALITY IN GROUPS BY Roger D. Konyndyk A group G is residually central if each non—identity element 9 E G is not an element of [g,G]. An unsolved problem is whether such groups must have a central series. Some partial solutions to this problem are obtained. Resid— ually central groups in which each element has only finitely many conjugates are locally nilpotent and therefore have a central series. Using group ring techniques, it is shown that a finitely generated residually central abelian by nil— potent group is residually nilpotent. The question of when the standard wreath product W = Avan of groups A and G is residually central is also taken up. If A and G are locally nilpotent, then W is residually central if and only if either G is torsion-free or there exists a prime p such that all elements of W of finite order have p-power order. . _ G _ If g E G, define Ro(g) — g and Rn(g) - TRn_1(g),G] for positive integers n. If for each 9 6 G, D Rn(g) = l, . n:O then G is called a (*)—group. (*)-groups are ZD—groups whose lower central series has length at most w + l, where w is the first limit ordinal. Many classes of (*)-groups must be residually nilpotent: wreath products of non-trivial groups, Roger D. Konyndyk nilpotent by cyclic groups, and cyclic by nilpotent groups. Counterexamples show that property (*) is not equivalent to residual nilpotence. Let x be one of the following classes of groups: residually nilpotent groups, (*l-groups, residually solvable groups, residually finite groups. If G is a I-group, then G/H is an 1 group if H meets any of the following con— ditions: a) H is maximal with respect to H g G and Yn(H) = l, where n‘g 2. b) H is maximal with respect to H Q G and H(n) = 1, where n.) l. c) H G G and H = CG(K) for some subset. K of G. SPECIALIZATIONS OF RESIDUAL CENTRALITY IN GROUPS BY (“\U/ Roger DI Konyndyk A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1975 A C KN OWLED GMENT S I wish to thank Professor R.E. Phillips for his helpful suggestions and advice. I must also thank my wife for her encouragement and love. ii TABLE OF CONTENTS I . BASIC DEFINITIONS AND RESULTS . . . . . . . . . 1 III. RESIDUALLY CENTRAL GROUPS . . . . . . . . . . . 9 III. RESIDUALLY CENTRAL WREATH PRODUCTS . . . . . . 14 Iv. A SPECIALIZATION OF RESIDUAL CENTRALITY . . . . 26 v . DESCENDANCE AND HOMOMORPHIC IMAGES . . . . . . 41 BIBLIOGRAPHY . . . . . . . . ; . . . . . . . . 49 iii CHAPTER I BASIC DEFINITIONS AND RESULTS Let G be a group, 91,...,gn E G. Then the com— mutator of 91 and g2 is {91,92] = 9119319192, and the F n—fold commutator is 191,...,gn_1,gn] = [[gl’°"’gn-l]’gn] for n > 2. If A and B are subsets of G, then [A,B] = <[ a,b]:a E A and b 6 B:>. If a is an ordinal greater than 2, then [[A,BB],B] if a has a predecessor 5 [mas] = n{[A,BB]:B < a} if a is a limit ordinal. In the case A = {9}, it will be convenient to write [g,aB] for [lg},a3]. If in addition B = G, we set Ro(g) = gG, the normal Closure of g in G, and 'Ralg) = [9,OG] for ordinals a > O. The following identities are routine to verify and will be used repeatedly. Lemma 1.1. Let x,y,z be elements of a group G. Then [x,21[x,ylz (l) [X.y2l (2) [Xi/,2] ix,ZJY[y,ZJ- The next few results are corollaries of 1.1. Proofs may be found in [17, pp. 43-44]. Lemma 1.2. Let A,B be non—empty subsets of a group such that B is a subgroup. Then [A,B,B] g [A,B] and [A,B] <1 . Lemma 1.3 (The three subgroup lemma). Let A,B,C be sub- groups of a group G. If any two of the subgroups [A,B,C], [B,C,A], [C,A,B] are contained in a normal subgroup of G, then so is the third. The following definitions are taken from [17] and [18]. For a group G and ordinal a, define yl(G) = G [YB(G)’G] if a has a predecessor B Ya(G) = DIYB(G):B < a} if a is a limit ordinal. G is said to be nilpotent of class .gc: if there is an in— teger c such that yc+l(G) = 1. The subgroups ya(G) are characteristic in G and form the lower central series of G. There must be a first ordinal ab such that Yab(G) = Yab+l(G); this subgroup is called the hypocenter of G. If for some do, yab(G) = 1, then G is called a ZD-group (hypo- central). The elements of the upper central series of G are defined by l |-‘ gO(G) gl(G) = the center of G Qa(G)/QB(G) = §1(ZE%ET) if a has a predecessor B. QQKG) = U{CB(G):B < a} if a is a limit ordinal. There is a first ordinal do such that gab(G) = gab+1(G); this subgroup is called the hypercenter of G. If for some ordinal ab, gu (G) = G, then G is called a ZA-group (hyper— 0 central). Another characterization of ZA-groups may be found in [12, p. 219]: Theorem 1.4. G is a ZA—group if and only if for every sequence xl,x2,... of elements of G, there is an integer n such that [xl,x2,...,xn] = 1. A central series is a collection Q of subgroups {Ga:a E I} of G such that (i) I is a fully ordered set such that if a, B 6 I and a < B, then Ga-3 65' (ii) If Ga.< GB’ a,B E I, and no element of Q is properly between Ga and GB’ then [GB’G]-S G07 that is, Ga/GB g g1(G/GB). (iii) For any subset A of I, U[Ga:a E.A} E Q and fifGafiI E A} E Q. (iv) [1} es and G e e. A group which has a central series is called a Z-group; thus ZA- and ZD-groups are Z-groups. If P is a group-theoretic property, then the group G has P residually if for every 1 ¥ 9 6 G, there is a normal subgroup N9 of G such that g A'N and 9 G/Ng has P. It is an easy exercise to show that G is G) residually nilpotent if and only if yw(G) = n Yn(G) = l. (w denotes the first infinite ordinal.) FolIOiing Kurosh [12], a group G is said to have P locally if every finitely generated subgroup of G has the property P. P is called a local property if every group which has P locally itself has P. Mal'cev has shown that the pr0perty of being a Z—group is a local property. (For a proof, see [18, pp. 93- 99].) A group G is residually central if for all 1 # g e G, g E [g,G]. This is equivalent to the condition that for all 1 #'g 6 G, there is a normal subgroup N9 of G such that g £ Ng and gNg E g(G/Ng); i.e. every non-trivial element of G is residually in the center of G. Residually central groups were first studied by Durbin in [3] and [4]. They have also been discussed by Ayoub [1], Slotterbeck [21], and Stanley [22] and [23]. The following results may be found in [3] and [18, pp. 6-8]. Proposition 1.5. If G is residually central and satisfies Min-n (the minimal condition on normal subgroups), then G satisfies Min and is hypercentral. Proposition 1.6. G is residually central if any of the following conditions holds: (1) G is a subgroUp of a residually central group. (2) G is locally a residually central group. (3) G is residually a residually central group. Proposition 1.7. If G is finite, then G is residually central if and only if G is nilpotent. Proposition 1.8. Let G be a residually central group, and let N be a normal subgroup of G contained in the hyper- center of G. Then G/N is residually central. Proposition 1.9. Elements of a reSidually central group which have relatively prime, finite orders commute. Stanley in [23] has obtained more information concerning when homorphic images of residually central groups are residually central. In general, however, the question appears quite difficult and little progress has been made beyond Proposition 1.8. Residually central groups are discussed in Chapters II and III. In Chapter II, relationships between residual centrality and other group_theoretical properties are dis- cussed. The main result is Theorem 2.7: Let G be a finitely generated residually central group with a normal abelian sub- group A. If G/A is nilpotent, then G is residually nilpotent. In Chapter III we take up the question of when the standard restricted wreath product W = Avan is residually central. The question is completely answered in the case where G is orderable and in the case where A and G are locally nilpotent. If G is a Z-group with central series Q = [Gaia e I], and l ¥'g 6 G, then rflGa]g 6 Ga] is an ele— ment G of Q, and U[Ga[g 5 Ga] is an element G “I “2 of Q. No element Ga of Q can satisfy Ga < Go. < Ga . 2 1 Thus [Gal,G] g Gaz, and [g,G] g [Gal,G] g Gaz. Since 9 ('Ga , g £ [g,G]. Thus any Z-group is residually central. 2 Whether or not the converse is true is unknown. The close relationship between residually central groups and Z—groups may be seen in the following theorem due to Hickin and Phillips [10]. Proposition 1.10. G is a Z-group if and only if for every finitely generated subgroup 1 ¥ K of G, Kié [K,G]. In Chapter IV a specialization of residual centrality is examined: A group G is a (*)—group if for all g e G, % Rn(g) = 1. It turns out that property .(*) is very closetx> iegidual nilpotence: A (*)-group is a ZD-group whose lower central series has length at most w + 1 (Proposition 4.4). Property (*) turns out to be equivalent to residual nil- potence for wreath products of groups (Theorem 4.13), nil- potent by cyclic groups (Proposition 4.15), and groups with trivial center (Corollary 4.6). Chapter V looks at homomorphic images of (*)-groups and other classes of groups, and looks at descendance in (*)—groups. We obtain Corollary 5.4. Let I be one of the following Classes of groups: Residually nilpotent groups, residually solvable groups, residually finite groups. If G is an x-group, then G/H is an z—group if H satisfies any of the follow- ing conditions: (1) ‘H is maximal with respect to H <16 and H(n) th H(n) - denotes the n derived subgroup — l, where of H, nig 1. (2) H is maximal with respect to H 4 G and yn+1(H) = l, where n.2 l. (3) H G G and H = CG(K) for some subset K of G. Corollary 5.9. Let G be a (*)-group, H as in (1), (2), or (3) of 5.4. Then G/H is residually nilpotent. Results on descendance are found in 5.10: If G is a (*)- group, and A is an abelian subgroup of G or if Aug gw(G), then A is descendant in G of order type at most w + 1. We close this section by stating two lemmas for future use . Lemma 1.11. If G is any group and g e G, then [g,G] = G . [g ,G]. Hence If n 2 o, Rn+l(9) = [Rn(g), G]. Proof: Since gG = <9 > [g,G], [96,6] [<9 > [9.61.61 "A [<9>,G][9’G][[9,G],G] (by 1.1) [<9 >,G][9,G,G] (since [,G] <1G) HA [<9>.G][9,G] (by 1.2) Thus it suffices to show that for any positive integer n and element h 6 G, [gn,h] 6 [9,6] and [g-n,h] E [g,G]. But [9“.h] = [ggn’1.h1 = [g,hlgn-l[9n-l,h] e [9.6] by in- duction on n. Similarly, [g_n,h] 6 [g_l,G]. Now [g-l,h]g = 9-1 9 h-l l [9,6]- g‘ hg = [7mg] e [9.6]. Since [9.6] <16, [9'1,G] 3 Lemma 1.12. If 91,...,gn 6 G, then % [gi,G] = i=1 [’ G] <1 C" Proof: Since eaCh [gi,G]_g [,cfl is the product of finitely many elements of +1 the form [gi ..... gi ,h]", where h.€G, and 1.3 ij g_n. But 1 m g. ...g. l 1 2 m [g (9- ----- 9- ),h] = [9 ,h] [9- ---~g- ,hl = 11 12 1m 11 12 1m gi2 ..... gi gi n [91 ,h] m ...[9i ,h] m[gi ,h] e W [91,6], since 1 m—l m i=1 each [gi,G] <1 G. Finally [, G] is the product of finitely many normal subgroups of G and therefore is normal. CHAPTER II RESIDUALLY CENTRAL GROUPS In this chapter relationships between residually central groups and other Classes of groups are examined. An FC—group is a group in which each element has only finitely many conjugates. Proposition 2.1. A residually central FC—group is locally nilpotent. Proof: Let G be a residually central FC—group, and H = a finitely generated subgroup. It suffices to show that H must be nilpotent. By 1.8, H/gl(H) is residually central. Since each hi has only finitely many conjugates in H, each [H:CH(hi)] < a. gl(H) = iEHCH(hi) is the intersection of finitely many subgroups each of finite index in H and therefore has finite index in H. Thus H/§1(H) is a finite residually central group, which must be nilpotent, by 1.7. Then H itself is nilpotent. Proposition 2.2. Let G be a residually central group with a normal subgroup N suCh that G/N is nilpotent and N satisfies the minimal condition on subgroups. Then G is a ZA—group. 10 Proof: Let xl,x2,... be any set of elements of G, and let yk = [X1,x2,...,xk]. By 1.4, it suffices to show that some yk = 1. Since G/N is nilpotent, there is an integer n such that Yn(G) g.N, and so yn E N. Let i be a non- negative integer. If y = 1, there is nothing to prove. n+i If yn+i # 1, then yn+i é [yn+i’G]’ since G is residually central. yn+i+l = [Yn+i’xn+i+l] E [yn+i’G] yfi+l > ... of F. Since F G G satisfies Min, for some j we must have yn+j = yn+j+l° This can happen only if yn+j = 1. Corollary 2.3. Let G be a finitely generated residually central group. If G has a finite normal subgroup N such that G/N is nilpotent, then G is nilpotent. Proof: Since N is finite, N satisfies Min, and 2.2 applies. Mal'cev [17, p. 50] has shown that ZA-groups are locally nil- potent, completing the proof. The necessity of the condition in 2.2 that the subgroup N has Min is shown by the example of the infinite dihedral group. This group has two generators, is metabelian, and is residually nilpotent (hence residually central), but is not nilpotent. If the group were a ZA-group, it would be locally nilpotent and hence nilpotent. If G is a group with a normal abelian subgroup A, and F = G/A, then A may be viewed as a right ZF-module, 11 where Z? is the integral group ring. Whenever this is done, A will be written additively and P will be written n multiplicatively. Thus if a E A and Z) zi(giA) e 2?, 1=1 where 2i 6 Z, 9i 6 G, 1.3 1.3 n, then the module action n a( Z) zi(giA)) is defined to be (in multiplicative notation) i=1 n zi 91 . . W (a ) ; the module multiplication lS basically that of i=1 group conjugation. It is routine to check that this is well— defined. Note that submodules are subgroups of A which are normal in G. Note also that for a e A,-g E G, [a,g] = a-lag may be written in module notation as —a + a(gA) = a(gA-—1). The augmentation ideal A of zr is defined to be the (two- sided) ideal generated by {y-—1]Y E F}. .If B is any subset of A, then [B,G] may be written as BA in module notation; repeating this m times shows that [B,mG] = BAm for positive integers m. A may be characterized by the well-known Lemma 2.4. If F is a group and Z? the integral group n ring, then the augmentation ideal A = [ Z) ziYi‘zi 6 Z, n i-l Yi E T, and Z) zi = 0]. i=1 Proof: The containment : is clear. Conversely, suppose n n n that z.Y. E Z? with Z) z. = 0. Then. 2) z.Y. - O = i=1 1 1 i=1 1 i=1 1 1 n n = z. = .Z) zi(Yi-l) E A. ll . 1 1 1:1 i=1 1 Let R be a ring and I an ideal in R. Following Roseblade [20], I is said to be a polycentral ideal of R if there is a Chain 0 = 10.3 11.5 ... g IC = I of ideals of 12 R such that I /Ii is generated by one element in the 1+1 center of R/Ii, 0.3 i < c. Roseblade has proved [20] Theorem 2.5. Every ideal of the integral group ring ZF is polycentral if and only if F is a finitely generated nilpotent group. Robinson in [19] uses the term polycentral to mean that each I /Ii is generated by some subset of the 1+1 center of R. He shows, in Theorem 5 of a preprint of [19], Theorem 2.6. Let R be a ring, I a polycentral ideal of R, and M a noetherian R—module. Then a 6 3 MIn n=1 if and only if a = ai for some i E I. These two results give Theorem 2.7. Let G be a finitely generated residually central group with a normal abelian subgroup A. If G/A is nilpotent, then G is residually nilpotent. Proof: For some integer n, yn(G).g.A since G/A is nil- potent. Let A1 = yn(G). By [2, Theorem 3.6], if G = <, then A1 = <[911’g12"°"91n]‘l'g lj.s k:> G/A1 is nilpotent, and A1 is finitely generated as a zremodule, Where T = G/Al. Now G is residually nilpotent if and only if n [A1,mG] = 1. If A is the augmentation m=1 an ideal of zr this condition is equivalent to n AlAm = o m=l in module notation. By 2.5, A is a polycentral ideal of 13 Zr in the senses of both Roseblade and Robinson. P. Hall has shown [7, Theorem 1] that Z? is a right-noetherian ring. Since Al is a finitely generated right zrbmodule, A1 is a noetherian module. By 2.6, a 6 fil AlAm if and only if a = a5 for some 5 e A. This mggns that a E aA, which says that a e [a,G]. Thus a must be the identity, since G is residually central. Corollary 2.8. Suppose G is a semi-direct product of A by F, where A is abelian and F is finite. If G is finitely generated and residually central, then G is re— sidually nilpotent. Proof: Since F is finite and residually central, F is nilpotent, and 2.7 applies. Phillips and Roseblade recently have shown [15] that if in 2.8 G is merely an extension of A by F instead of a semi—direct product, then G is a ZD-group, and yw+d(G) = 1 for some integer d. CHAPTER III RESIDUALLY CENTRAL WREATH PRODUCTS Let A and G be non-trivial groups. For each 9 E G, let Ag =.A, and set A =,LKAg|g 6 G}. Any element a of A can be thought of a function acG q.A such that l “(9) = for all but finitely many 9 e G. Map G into 1 Aut(A) by dh(g) = a(gh- ), for g, h 6 G, a E A. The re- sulting semi-direct product W = A]G is called the (standard restricted) wreath product of A by G, written W = Avan. The subgroup A is called the base group. If a 6 A, we shall usually write a = E) aigi to mean that a(gi) = ai, 1.5 1.3 n, and a(g) = l 1;; g t [91,...,gn]. In this n notation, if g E G, then 09 = Z) aigig. If B G.A, then 1=1 AwrG is a homomorphic image of AwrG in the obvious way: B the kernel of the homomorphism is A = BG = LHBg'g e G]. More information about the structure of W may be found in P. Neumann's paper [13]. Hartley [9] has determined which wreath products are residually nilpotent. Motivated by this, we now turn to the question of which wreath products are residually cen— tal. In the sequel, W will always denote the wreath product Arer, and A will denote the base group. Note that A and 14 15- G can be embedded in W. Since subgroups of residually central groups are residually central, A and G must be residually central if W is to be. Proposition 3.1. Suppose that W = Axer is residually central. If G is infinite, then A is a Z-group. Proof: Let al,...,an e A, K e . By 1.10, it suffices to show that K g [K,A] . Let gl, .. .,gn be . n . _ distinct elements of G, and let a = Z a. 1 E A. Since _ I . . 1’1 - .11 gi W 1s reSIdually central, 0 A [a,w].2 [a,A] = L,[ai,A] 6 i=1 as a direct sum. Fix 9 E G, and let bi [ai,A], l‘g iig n. Then hi 1 e [ai,A] 1 _<_ [a,w] <1w; thus bi = (bi 1) 1 g e n [a,w]. Hence ( 7r [ai,A])g = [K,A]9 _<_ [a,w] by 1.11. Since i=1 9 was arbitrary, Z)[[K,A]g:g 6 G} g [a,W]. If K g [K,A], n g. n g. 1 6.2[181111 s then aie[K,A], 1_<_ign, and a: 23 a. i=1 i=1 1 [a,W], a contradiction. A group G is ordered if there is a total order < on G such that if a < b in G and c,d e G, then cad < cbd: that is, the order on G is preserved by right and left multiplication. A group on which it is possible to impose such an order is called orderable. Every orderable group G must be torsion—free: If 1 # g e G, then either 9 > 1 or g-1 > 1. An easy induction then shows that gn > 1 or (g_1)n > 1 for every positive integer n. It is also clear that subgroups of orderable groups are orderable. For further 16 information, the reader may consult [11]. The following facts are proved there. Proposition 3.2.[11, pp. 4,5] The cartesian product of orderable groups is an orderable group, and any group which is residually an orderable group is orderable. Proposition 3.3.[11, p. 10] A group is orderable if and only if every finitely generated subgroup is orderable: that is, orderability is a local property. Proposition 3.4.[ll, p. 16] A locally nilpotent torsion-free group is orderable. Proposition 3.5.[11, p. 17] All free groups are orderable. If g is an element of the ordered group G, let ‘9‘ = max[g,g—1}. Then 9 is said to be infinitely small relative to h, written 9 << h, if |g|n < [h] for all positive in- tegers n. A subset K of G is convex if for all g 6 G, h 6 K, lg‘ < [h] implies that g 6 K. The next result is in [11, p. 14,15]. Proposition 3.6. Let G .be an ordered group, g,h E G. Then (1) 9 << h if and only if there is a convex subgroup containing 9 but not h. (2) l[g,h]I << max{]g[,‘h[}. (3) If G is nilpotent, then [[g,h]] << ‘9]. 17 Lemma 3.7. If G is a Z-group with central series {Ga[a E I, I a totally ordered set}, then G is orderable if G/Ga is torsion-free, for all a e I. Proof: This is really a corollary of Theorem 2.2.3 in [11, p. 16]. Each GQL<1G’ and if a,B e I such that Ga.< GB and there is no Y e I for which Ga < G < GB, then [G5,G] Y .g Ga’ and GB/Ga is a torsion-free abelian group, and hence is orderable, by 3.4. Since GB/Gaflg gl(G/Ga), elements of GB/Ga. are fixed under conjugation by elements of G/Ga, Thus {Gaia E I} meets the conditions of Theorem 2.2.3 of [11]. Lemma 3.8. Let A and G be residually central groups. Then W = Avan is residually central if for all 1 # a e A, a g [a,G][a.A]G. Proof: Since W is the semi-direct product AG, any element of W can be written uniquely in the form 0g, where a 6 A and g 6 G. If g # 1, then 09 A A[g,G], since g £ [g,G]. Now [09.17] s [a,w][g.W] g Herbie] s itg,G][g,A]G s A[9:Gl- Thus 09 £ [ag,W]. If g = 1, then [a,W] = [a,AG] g [a,G][a,A]G. Thus W is residually central if 0 £ [a,G][a,A]G. Theorem 3.9. If G is a residually central ordered group, and A is a Z-group, then W = Avan is residually central. m 9' _ Proof: Let a: E, aileA, where giEG, aiEA, l_<_ i g’m. By 3.8, it is enough to assume that a E [a,G][a,A]G I.‘ 18 and reach a contradiction. Let L = [<,.A]. Since A is a Z-group, some ai A L, by 1.10. If L = ZKLg:g 6 G} 1 or hl < l, or both. (Note that n.2 2, since if, zj = 0.) We treat the case where hn > 1: the case h1 < l iglalmost identical. First, suppose that some ai has infinite order. Let gig = max[gi[ai has 1nf1n1te order}. Then giohnj>gi0 in c, 19 th Since a = a6, the element in the 91 h;— "slot" of a, 0 say b, must be equal to Z) z.a.g.h.. By definition, .h = g h 3 1 1 3 91 j iO n b has finite order. However, znai has infinite order, 0 and no other summand of b has infinite order. Thus b has infinite order, a contradiction. Next, suppose that A3 ==<:a1,...,anl> is a finite p—group, for some prime p. If for some = O, 30’ zjoai lgigm, let 5’: szhj. Then a5’=a5=a; we may mo therefore assume that for each j there is an i such that zjai # O. S1nce gm > gi 1f 1 < m and hn > hj 1f 3 < n, . . . th . . gihj < gmhn 1f 1 < m or 3 < n. Thus the gmhn p051t1on in G5 is znam. Because 9mhn > gi, 1‘3 1 g_m, and a = a6, znam = O. For some i, znai ¥ 0: not all the elements of A3 = can have the same order, and hence A3 has exponent greater than p. Suppose the exponent of A3 is pk. Let A4 = A3/pk-1A3, and let G be the natural map. 0 # ¢(a), and ¢(a) = ¢(a5) = ¢(a)5. Since A4 has exponent p, this is impossible. Finally, suppose that A is a finite abelian group. 3 Then there are primes p1,...,pr such that A3 = 314:..49Br, where B is apkfl-QI‘OUP: 1 _<_ k S. r. AS before, 191: A5 = k A3/(BZG>..AaBr) with Y the natural map. Then 0 ¥ 1(a) = 1(05) = Y(G)6 in the pl-group A5, ‘which is impossible. 20 This shows that if G is a residually central order- able group, then W is residually central if and only if A is a Z-group. Free groups are orderable (3.5) and are re- sidually nilpotent [18, Theorem 9.11], and hence are Z-groups. Thus the wreath product of two free groups is residually cen- tral. Lemma 3.10. Suppose that W = Avan is residually central, and G has an element 9 of prime order p. Then every element of A and of G of finite order has p-power order. Proof: Suppose a E A has prime order q # p. Identify a with the element of the base group A ,defined by a(lG) = a A is impossible. and a(h) = 1 if h ¥ 16. By 1.9, a and g commute, which Suppose h E G has prime order q ¥ p. By 1.9, h and g commute, so that (they) is cyclic of order pq. Let 1 74 a e A. Wl = wr is a finitely generated, met- abelian, residually central group. By 2.7, W1 is residually nilpotent. However, Hartley [9] has shown that this is im- possible. Theorem 3.11. Suppose that A and G are locally nilpotent. Then W = Avan is residually central if and only if either (1) G is torsion-free, or (2) All elements of G finite order have p—power order, where p is a prime, and all elements of A of finite order also have p-power order. 21 Proof: The necessity of (l) or (2) is clear from Lemma 3.10. If (1) holds, then by 3.4, G is orderable, and Theorem 3.9 applies. Suppose (2) holds. Since residual centrality is a local property, it suffices to consider a finitely generated subgroup of W. Bash :1 = Gigi , where _ _ ij ai e A and 9i 6 G. Each (11 - j§l aij . Hence Thus we may assume that A and G are finitely generated and therefore are nilpotent. By [2, Theorem 2.1], A can be em— bedded in PAGTA, and G can be embedded in PGQTG, where PA’PG are finite p-groups, and TA’TG are torsion—free fi— nitely generated nilpotent groups. By Lemma 3.8, it suffices to show that if a = Z“, akgk 6 A, then a A [a,G][a,A]G. Suppose that there is an a.k:tch that a e [a,G][a,A]G. Since A ,is nilpotent, there is an Integer r such that each ai 6 gr(A) and some ai é gr_l(A). Then [a,il g [,A]G A [gr(A).A]G IA gr_1(A)G. IA Wl= (A/gr_1(A)) wrG isa homomorphic image of W in the obvious way. Let 3 denote the image of a in W1. Because a e [a,G][a»A]G, a 6 [a,G][a,A/gr_l(A)]G = [B,G] in W1. Let 22 A1 = gr(A)/gr_l(A). Then Alvan is a subgroup of Wl which contains 6. Also, [B,G] 3 A1, since a E QILA/Cr_l(A)), a characteristic subgroup of A/§r_l(A). By [2, Corollary 2.11], every element of A1 of finite order has p-power order. By [2, Theorem 2.2], A1 and G are residually finite p—groups, and hence are residually nilpotent p—groups of finite exponent. Since a e [B,G], Alvan is nOt residually central, and there- fore not residually nilpotent. However, Hartley [9] has shown that Alrer is residually nilpotent, a contradiction. Corollary 3.12. If A is abelian and G is locally nilpotent, then W = Avan is residually central if and only if W is locally a residually nilpotent group. Proof: The sufficiency of the Condition is clear. Theorem 3.11 and Theorems Bl and B2 of [9] combine to prove the necessity. Thus we have succeeded in classifying those restricted wreath products W = Axer which are residually central in the case where G is orderable and in the case where A [and G are locally nilpotent. In addition to this, Hartley's paper [9] gives conditions for W to be residually nilpotent: his condi- tions clearly are sufficient conditions for W to be residually central. Necessary conditions are that A must be a Z-group if G is infinite, G can have at most one relevant prime, and if G has an element of prime order p, then every element of A and of G of finite order has p—power order. To expand our results to the case where A is not locally nilpotent appears to be difficult. In order to use 23 group ring techniques, as was done above, and as Hartley did in [9], it is necessary to work within an abelian "slice" of A and to know something about the orders of elements of that slice. This is much more difficult if A is not locally nilpotent. Thus it seems likely that different techniques will be required to expand 3.11 significantly. If the usual base group E{Ag|g E G} is replaced by W[Ag|g 6 G], the resulting group is called the unrestricted wreath product, denoted by AthsG. Because this is a much "larger" group, one would expect that far more restrictive conditions would be necessary to make Aler residually central. That this is indeed the case is illustrated by the following result. Proposition 3.13. Let W = AVhrG. If G contains an element 9 of infinite order, and A contains an element a1 of finite order or a 2-divisible subgroup, then W is not resid— ually central. Proof: Suppose A contains a 2-divisible subgroup A1. It is enough to show that A1Wr is not residually central. Let 1 ¥ a 6 A1. For each positive integer i, there . 21_ _2 . Is an element bi such that bi — a and bi - bi+l° Define a e W{A1 n‘n E Z] by 9 2i a if 1.2 0 b. if i < O. 24 —1 i —1 g'1 i Then [0,9 1(9 ) = (a a )(9 ) 1 1 +1 = (a(91))‘ 0(9 g ) 1 1+1) (a(gi))' -1‘ 2 a 0(9 i+1 if we denote a(gi). Hence [a,g-l] = a, and W is not residually central. Suppose A has an element a1 of finite order. If the order of al is odd, then A contains a subgroup A isomorphic to a cyclic group odd prime order. Such a 1 group is 2-divisible, and the above argument applies. Suppose, A contains an element a1 of order two. Again, it suffices to show that Wr (g > is not re— sidually central. Define a in the base group by a if 3 does not divide i i 0(9 ) = 1 if i is divisible by 3. Then [a,g‘1](g1> = (a(g1))‘1a(g1+1> = a(g1)a(g1+l), since a = a-1 {:a if i or i+1 is divisible by 3 1 otherwise. 25 . ' _ . . .+ Slm11arly, [0,9 21(91) = a(91)a(g1 2) {:a if i or i+2 is divisible by 3 1 otherwise. -1 [2 i 1 if i is divisible by 3 Then ([a,9 lia,g 1(9 ) = 8 otherwise. Therefore [a,g“ a, a 6 [0,6], and W is not residually central. CHAPTER IV A SPECIALIZATION OF RESIDUAL CENTRALITY Recall that, for g E G, Ro(g) = 96 [Rn(g),G] for non—negative integers n. If for each element 9 of a group G, A Rn(g) = 1, then G is said to be a (*)- n=0 . group. Because each Rn(g) g Yn+1(G), every residually nil- potent group is a (*)—group. Note also that if g 6 [g,G], a then 111(9) = [g,G] = [g,G:G] = R2(g). Thus if n Rn(9) =1: n=0 then [g,G] > [g,G,G], and therefore every (*)-group is re- sidually central. By 1.7, property (*) is equivalent to nilpotence for finite groups. Proposition 4.1. The Class of (*)-groups is closed under the taking of subgroups and Cartesian products: hence a residually (*)-group is a (*)-group. Proof: Let H be a subgroup of G, and let h 6 H. Since G a for each n [h’nG1-2 [h,nH], 1 = n21[h’nG]'2 n21[h’nH]' Let {Gi|i E I} be‘a collection of (*)-group, and 1] .g N [91,61], and for 161 161 iEI each n, Rn(g) g N Rn(gi)' 161 Since n Rn(gi) = 1G for each i 6 I, A Rn(g) = 1. n=0 i n=0 26 27 Suppose a group G is residually a (*)-group. Then for each 9 6 G, there is a normal subgroup Hg of G such that g é'Hg and hg is a (*)-group. G can be embedded in the Cartesian product W[G/H:g E G} by the map 9 a (ng1xeG' The Cartesian product of the (*)-groups [G/Hg:g 6 G} is a (*)-group, and so G is a (*)-group. Free groups are residually nilpotent [18, p. 117] and thus are (*)-groups. Since every group is a homomorphic image of a free group, and, e.g., finite non-nilpotent groups are not (*)-groups, a homomorphic image of a (*)-group need not be a (*)-group. The symmetric group on three symbols is metabelian but not a (*)—group: this shows that a (*)-group extended by a (*)-group need not be a (*)-group. The following lemma is a fairly well-known extension of the three-subgroup Lemma 1.3. Lemma 4.2. Let H and K be subgroups of a group G such that [H,nK] < G for all positive integers n. Then for any such n, [Yn(K),H] S [H,nK]. Proof: Induct on n. The case n = l is clear. Suppose n.2 2. Then [H.nK] = [[H.(n_l)r<],r<1 2 [[Y(n_1)(K),H],Kl by induction = [Y(n_l) (K),HJK] ° Similarly, 28 [H,nK] =[[H,K],(n-1)K].2 [Y(n_1)(K),[H,K]] by induction = [H,K,y(n_l)(K)]. By the three subgroup Lemma 1.3, [H,nK].2 [K,Y(n_1)(K),H] = [Yn(K),H]- We begin by exploring the relationships between (*)-groups and some other classes of groups. Proposition 4.3. A group G is hypercentral with the upper central series having length at most T if and only if fOr each 1 # g 6 G, there is an integer ng such that R.n (9)==l. 9 In particular, such a group is a (*)—group. Q Proof: Necessity. Suppose that G = gw(G) = U gn(G), and n=l let 1 ¥ g e G. Let ng be the least integer such that g E gn (G). For each positive integer n, [gn(G),G] g gn_1(G), 9 and so Rn (9) = [9,n G].S [Qn (G),n (6)] S 90(G) = 1. 9 9 9 9 Sufficiency. Suppose that for all g E G, there is an integer n such that R (g) = 1. Since g (G) = 9 n9 n9 [x e G:[x,n (G)] = l} ,g e gn (c). Thus G = U gn(G). 9 9 n=1 However, a group can be hypercentral of length m + l and yet contain an element 9 such that Ra(g) # l for all ordinals a. 29 an Example 4.1. Let P be a 2 -group, and let a E Aut(P) be defined by pa = p-1, where p 6 P. Let G = P] m Dih(Z(2m)). Let p e P. [a,p] = (p_l)ap = p2. Because P is divisible, P2 = P; i.e., every element of P is the square of some other element of P. Hence [a,P] = P2 = P, and Ra(a) = P for every ordinal a.2 1. To see that G has ZA-length w + 1, view P as the abelian group generated by {pi:i = 1,2,3,...}, where pi = l and p? 1+1 = pi for i 2,1. Suppose that piaJ 6 gl(G), where i 2 1, j = 0 or 1. (piaj)‘1l = pJT-l a:I = piaJ if and only if i = 1, for only p1 = p11. Also, pzpla = p3 = p31. Thus gl(G) = [1,pl}. Note that G/gl(G) e—G, and so g2(G)= <:p2:>. Similarly, gn(G) = for each integer n. Then §(G)=UC(G)=U

=P. “1 n=1n n=1 ’1 Slnce G/P ls abelian, gw+l(G) = G. Proposition 4.4. A (*)—group is a ZD-group whose lower central series has length at most w + 1. Proof: Let G be a (*)—group, and let 9 E G. By Lemma m CD 3.2, n [g,Yn(G)] _<_ n Rn(g) = 1. Thus n=1 n=1 [gm (6)] = [9, 7% Y (on _<_ 312(9): 1. w n=1 n n=1 n Thus G ntr 1' G. Th G Yw( ) ce a lzes every 9 6 en Yw+1( ) [Yw(G),G] _g [gl(G),G] = 1. 30 Combining this with 4.3 gives a result proved by Hartley [8]. Corollary 4.5. A ZA-group of length at most w is also a ZD-group of length at most w + 1. Corollary 4.6. If G is a (*)-group, and g1(G) = 1, then G is residually nilpotent. Corollary 4.7. A (*)-group satisfying the minimal condition on normal subgroups is nilpotent. Corollary 4.8. A group G is residually nilpotent if and only if G is a (*)-group and for all 1 fl 9 e g1(G), there is a normal subgroup Mg of G such that 9 AIM and 9 G/Mg is nilpotent. Proof: The necessity of the condition is clear. Let 1 # g 6 G. To show that G is residually nilpotent, it is ne- Cessary to find a normal subgroup Mg of G such that g A'Mg and G/Mg is nilpotent. If g e g1(G), this is true by hypothesis. Suppose g £ C1(G). By 4.4, nEEYn(G).S 91(3), and so there is an integer n such that g A Yn(G). Since G/yn(G) is nilpotent, the result follows. An obvious question now is whether prOperty (*) is equivalent to residual nilpotence (ZD-length at most w) or to having ZD-length at most w+-l. The next two examples show that it is equivalent to neither. 31 Example 4.2. Let Dn = , the dihedral group of order 2n. It is easy to n-2 2 check that §1(Dn) = (b!1 > 'has order two and that Dn/g1(Dn) I'Dn-l if n.2 3. S1nce the nilpotency class of D is 2 (i.e., g2(D3) = D and g2_l(D3) ¥'D3), the 3 3 nilpotency class of Dn is n-l if n‘z 3. G Let G = Z)l%1, a group which has ZA-length w and ZD—length w. Let em nzglwm) 4 gl(Dn) by (5mmm ) = ’ bn , for m,n 2 3. Let H = , and set F = G/H, called the central product of the Dn's. Let L = g1(D3). If m ¥ n, then HDn/H n HDm/H = LH/H =- L/(H n L) = L. Thus §1(F) = LH/H has order two. Since G has ZA-length w, P has ZA-length Iggn and hence is a (*)-group. To show that r is not residually nilpotent, the following well-known lemma is needed. Lemma 4.9. If r is a ZA-group, then every non-trivial normal subgroup of F intersects §1(F) non-trivially. Proof: Suppose l #'N G r .and N n gl(r) = 1. Since r is a ZA-group, there is a least ordinal a such that IN 0 §a(P) > 1. Clearly, a cannot be a limit ordinal. Then a ‘has a predecessor B, and N n QB(F) = 1. Since Nl1g1(r) = 1, 1 3“ [N ngamm] _<_ N n[ga(1‘),r] _<_ N ngBU‘), a contradic- tion. 32 Since gl(P) has order two, every nontrivial normal subgroup of F contains g1(r). Since F contains c0pies of each Dn’ F cannot have finite nilpotency Class; no Yn(F) can be the identity. Thus for each n, €l(r)-S Yn(F), and 1#c(r)_g Fix/(1‘). 1 , n=1 n Glugkov [6] has constructed a very similar class of examples. He takes the central product of upper n.xn. uni— triangular matrices over a finite field K. For n.2 2, the center of the group Mn of upper unitriangular n xn matrices turns out to be [i o ...o a] 0 l 0 ... 0 0 ° a E K , 0 0 .......... l L. .J which is isomorphic to the additive group K. Each M.n is a nilpotent group, and so the central product has ZA-length .3 mo Let p be the Characteristic of K, and pp the order of K. If the additive group K is generated by k, then every non—trivial subgroup of K must contain (kw-.1 > , which is cyclic of order p. If w; is the central product of the Mn's, by 3.8 every non-trivial normal subgroup of m intersects CICM) e K non-trivially, and therefore must contain 33 .- H m—l 1 o o o kp o 1 o ...... 0 _o ......... o l - This subgroup is contained by Yn(W) for n'2 l, and is co therefore contained in n Ynom). Thus m is not residu- n=1 ally nilpotent. The next example shows that property (*) is not equivalent to having ZD-length at most w + 1. Example 4.3. Let H be the abelian group with presentation 1 <. (For a note on the exis- tence of such a group see [5, p. 118 and Theorem 36.1, p. 121].) Define a 6 Aut(H) by ha = h_1, h E H, and set G = H]. Y2(G) = [H, H] = [H,]. If h EH, then [h,a] = h-lha = hm2 , and thus Y2(G) = H2. Now sup- i-l pose that Yi(G) = H2 . Then i-l i-l i-l i 2 Yi+1(G)=[H2 ,H] = [H2 ,] = (H2 > = H2 Therefore a , 2i-l Y,,(G)= nYi(G)= nH =, i=2 i=2 and Ywflm) = [Yw(G),H] = [,H] = [,] = 1. However, Y2(G) = [H,] = R1(a), and so 34 Yi+l(G) = Ri(a) for 1.2 l. a Thus n R (a) = Y (G) = #11: G is not a (*)-group n=1 n w even though G is a ZD-group of length w + 1. All (*)-groups have ZD-length at most w + 1. The interesting question now is when (*)-groups have ZD—length m: that is, which (*)-groups are residually nilpotent. We first investigate (restricted) wreath products. Lemma 4.10. Let W = Avan, where G is infinite. If W has property (*), then W is residually nilpotent. Proof: By 4.6 it is sufficient to show that C1(W) = 1. However, this is already well-known: see [13, p. 34]. Now it suffices to consider W = Axer for finite groups G. If W is to be a (*)—group, both A and G must be (*)-groups, because both A and G are embedded in W. Then the finite group G must be nilpotent, since G is a ZD—group. If W is to be a (*)-group, ‘W must be- residually central; Lemma 3.10 now shows that G must be a p-group for some prime p. Since for some finite c, YC(G) = l, YC(W) is contained in the base group A: thus W is residually nilpotent if and only if FILA,fiW] = 1. If A is abelian, this reduces to the condIfion that n:1[A,nG] = 1. As in the discussion of residually central wreath products, A will be viewed as a module of the in— tegral group ring ZG in this case. 35 Let g 6 G have order p, and suppose that A is abelian. Let W* = Awr , and denote its base group by A7; If n'2 1, then [137, 1nl = [313 g].(n_1)] g 131.1(9). If W is a (*)-group, so is ‘W*, and thus [A , ]_<_ FR(g)=1, 1 n n=1 n IMDB n and W* is residually nilpotent. By Lemma 8 of [9], A is residually a p~group of finite exponent. By [9, Theorem Bl], W is residually nilpotent. We have proved Lemma 4.11. If A is abelian and G is finite, then W = Axer is a (*)-group if and only if W is residually nil- potent. In this case, there is a prime p such that G is a p-group, and A is residually a p—group of finite exponent. Lemma 4.12. Suppose W = Arer' is a (*)-group, where G is finite. Then for some prime p, G is a p—group, and A is residually a nilpotent p-group of finite exponent. Since by Theorem B1 of [9] such a group is residually nilpotent, this lemma and Lemma 4.10 complete the proof of Theorem 4.13. The standard wreath product W = AwrG of non-trivial groups has property (*) if and only if W is residually nilpotent. 36 Proof of 4.12. Suppose that W is a (*)—group, and let 1 a! a 6 A. Then wrG is also a (*)-group. By 4.11 G is a p—group, and if a has finite order, then the order of a is a power of p. Let g E G have order p, and let A be the augmentation ideal of the group ring Z<§;>. What follows is an adaptation of Lemma 8 of [9]. Claim. If n is a positive integer, then there is an element rn of Z such that pn(1-g) = rn<1-g)"“1"‘1”'l (1). Proof of claim: By [9, Lemma 6], p(l-g) 6 AP. For n = 1, then, there is an element r1 6 Z<§3> such that p(l-g) = rl(l-g)p, since Ap is generated by (l-g)p. Now suppose that (1) holds for some n‘z l. n(p—l)+l Pn+l(l-g) p(pn(l—g)) = p-rn(l-g) rn(l-g)n(p_1)p(l-g) n(p-l) rn(l"9 r1(1-9)p by the case rl=]. (rnrl) (1 _ g)n(p-l)+1P l)(1__g)(n+1)(p---l)+1 = (rnr , as desired. Identify A with the subgroup of the base group given by [f:G 4 A|f(g) = 1A if g a! 16}. Write additively, and view the base group of wr as a Z-module. n [ap ,9] may be written in module notation as pna - (pna)g = a ~pn(l-g). By (1), there is an element rn e Z such n(p—l)+1 n(p—l)+1 that apn(l-g) = arn(l-g) E aA Thus 37 n [ap ,9] E [a:n(p_1)+1] s [<9>,A,n(p_1)<9>] _<_ Rn(P_1)+1(g) in w. (2). Define Am = <29“: 3 e Aj> for positive integers m. Let n A =Ap(AnY n Then n(p—l)+l(w))° pn [Anna] 5. [A ,g][A nY l(w),9] n(p-1)+ .3 Rn(p—l)+l(g) by 4.2. Thus [nAn,9]< n- 1 n [An ,9] < nnl R n(p—1)+1(g) = 1 n=1 since W is a (*)-group. However, for any 1 # b E A, [h,g] #11. Thus [ n .A n,g]= implies that A .An = 1. n=1 n=1 Since An contains the subgroup generated by all pa“— powers of elements of A, A/An has exponent dividing pn. Because An conta1ns Yn(p—1)+1(A)’ each A/An is also nilpotent. Finally, if 1 ¥ b E A, there is an integer n such that b é‘An’ since 0 An = l, and A/An is a nil- n=1 potent p—group of finite exponent. Thus the lemma is estab- lished. Lemma 4.14. Suppose that the group G has a normal subgroup B such that G/B = for some a 6 G. Then for each positive integer n, Yn(G) = yn(B)Rn_1(a). 38 Proof: Induct on n. For n = l, Yl(B)RO(a) = 33 = G = Y1(G). Suppose that the lemma is true for some n.2 1. Then Yn+l(G) = [Yn(G),G] = [Yn(B)R(n_1)(a):G] V\ [Yn(B):G][R(n_1)(a):G] [Yn(_B),B <61 > JRn(a) M Hum). ][Yn(B),B] <3 >linen) S'Yn+l(B)Rn(a) by 4.2. The opposite inclusion is trivial. Proposition 4.15. A (*)-group G which is nilpotent by cyclic is residually nilpotent. Proof: There is a subgroup B <1 G such that G/B = for some a E G, and YC(B) = l for some integer C. For n.) c, by Lemma 4.14. Yn(G) = Yn(B)Rn_1(a) = Rn_l(a). a: Q Thus n Yn(G) = n R (a) = 1. n=1 n n-l 1 Proposition 4.16. A (*)-group G which is cyclic by nil- potent is residually nilpOtent. 39 Proof: There is an element a 6 G such that (21> Q G and G/ is nilpotent. Thus for some integer c, YC(G) g , and so for n2 0, Yc+n(G) g [,nG] = Rn(a). a: (G) g n Rn(a) = 1. Note that if <21) is finite, n=0 0 Y + n=0 C n then G is nilpotent. Proposition 4.17. Suppose that G has a normal subgroup B such that G/B is nilpotent, and B satisfies the minimal condition on normal subgroups. If G is a ZD—group, then G is nilpotent. Proof: For some integer C, YC(G)'S B. The subgroups YC(G).2 Yc+l(G)'2 --- form a decreasing chain in B. Since B satisfies Min-n, there is an integer n such that (G)== Yc+n Yc+n+1(G)' Slnce G 13 a ZD-group, Yc+n(G) = 1. In example 4.2, g1(r) is cyclic and T/§l(F) a D 3 residually nilpotent, this shows that the hypotheses of the n’ a residually nilpotent group. Since P is not Abfle last two propositions cannot be weakened to read, "G/ of G/B is residually nilpotent". Proposition 4.18. Let G be a (*)-group with a normal sub- group H such that gl(G) n H = l and G/H is residually nilpotent. Then G is residually nilpotent. Q Proof: Since G/H is residually nilpotent, Yw(G) = nlyn(G) n: 13 H. By 4.4, l = Yw+l(G) = [y (G),G]. Thus Yw(G) S c1 k. Then _ _ . . -l _ -1 _ hknk — x — hjnj’ 1mply1ng that hj hk — njnk 6 HtfiNk - 1. Thus if j > k, nj = nk'; nk E Nj for all j > k. But then nk E iLHNi = 1. Thus x = hk 6 H. Corollary 5.6. Let H be a normal subgroup of G such that H satisfies Min-n, and let y be one of the following Classes of groups: residually nilpotent groups, (*)—groups, residually solvable groups, countable residually finite groups. If G is a y-group, then G/H is a y-group. Corollary 5.4 can be improved in the case that z is the Class of (*)-groups. Proposition 5.7. Let G be a (*)-group, K any subset of G. Then n21 CG(K)Yn(G) = CG(K) . 45 Q Proof: Let k e K, x e n CG(K)Yn(G). For each n, x n=1 can be written as x = Cngn’ where Cn E CG(K) and 9n 6 Yn(G). Furthermore, [x,k] = [cn9n,k] = [gn,k] 6 [Yn(G),k] .3 Rn(k) by 4.2. Hence [x,k] E A Rn(k) = l, forcing x to centralize k. n=1 Corollary 5.8. Let G be a (*)-group. For any positive integer n, G/gn(G) is residually nilpotent. Proof: Set K = G in 5.7 and induct on n. Corollary 5.9. Let G be a (*)-group, n a positive in- teger. Then G/H is residually nilpotent if any one of the following conditions holds: (a) H is maximal with respect to H 4 G and H(n) = 1. (b) H is maximal with respect to H <16 and Yn+l(H) = 1' (C) H = CG(K), where K R1(g) 2 R2(g) Z Q is a normal series, and n (g > Rn(g)= (g > by Proposition n=1 5.5. (3) Consider the series G=Y(G)H2Y(G)H2°°° nY(G)H2H. l 2 n=1 n As in (1), each Yn+1(G)H < Yn(G)H. It remains to Show that a Q h <1 n Yn(G)H. Let h e H. Since H _<_ u gi(G), h 6 gm(G) n=1 i=1 ' O for some integer m. Thus Rm(h) = 1. Let y E n Yn(G)H. n=1 Then y can be written as y = g’h", where g’ 6 yn(G) and h E H. Thus 11" = h[h,y] = hth.g'h'] = hth.h'][h.g'][[h.g'].h'1. Now [h,g’] E [h,Ym(G)] S Rm(h) = l by 4.2. Hence hy = h[h,h’] E H as desired. BIBLIOGRAPHY [ l] [ 3] [4~] [ 5] [ 6] [ 7] [ 8] [10] [ll] [12] [13] BIBLIOGRAPHY Ayoub, C., 0n properties possessed by solvable and nilpotent groups, J. Austr. Math. Soc. 9 (1969), 218-227. Baumslag, 6., Lecture Notes on Nilpotent Groups, American Math. Soc. (Regional Conference Series in Mathematics, no. 2), Providence, Rhode Island, 1971. Durbin, J.R., Residually central elements in groups, J. Algebra 9 (1968), 408-413. , 0n nonmal factor coverings of groups, J. Algebra 12 (1969), 191-194. Fuchs, L., Abelian Groups, Pergamon Press, New York, 1960. Glugkov, V.M., 0n the central series of infinite groups (Russian), Mat. Sbornik N.S. 31 (73), 491- 496 (1952). Hall, P., Finiteness conditions for soluble groups, Proc. London Math. Soc. (3) 4 (1954), 419-436. Hartley, 8., The order-types of central series, Proc. Cambridge Philos. Soc. 61 (1965), 303-319. , The residual nilpotence of wreath products, Proc. London Math. SOC. (3) 20 (1970), 365-392. Hickin, K.K., and Phillips, R.E., 0n Classes of groups defined by systems of subgroups, Archiv der Math. 24 (1973), 346-350. Kokorin, A.I., and Kopytov, V.M., Fully Ordered Groups, transl. D. Louvish. John Wiley and Sons, Inc., New York, 1974. Kurosh, A.G., Theory of Groups, vol. II, transl. K.A. Hirsch. Chelsea, New York, 1956. Neuman, P., On the structure of standard wreath products of groups, Math. Z. 84 (1964), 343-373. 48 [14] [15] [l6] [17] [18] [19] [20] [21] [22] [23] 49 Petty, J.V., Weak homomorphic image Closed properties of series determined by Classes of groups. To appear. Phillips, R.E., and Roseblade, R.E., Zero—divisors in rings and centrality on groups, Notices of the AMS 22 (1975), A-398. Robinson, D.J.S., A property of the lower central series of a group, Math. Z. 107 (1968), 225-231. , Finiteness Conditions and Generalized Soluble Groups, Part I, Springer-Verlag, Berlin, 1972. , Finiteness Conditions and Generalized Soluble Groups, Part II, Springer-Verlag, Berlin, 1972. , Hypercentral ideals, noetherian modules, and a theorem of Stroud, J. Algebra 32 (1974), 234- 239. Roseblade, J.E., The integral group rings of hypercentral groups, Bull. London Math. Soc. 3 (1971), 351—355. Slotterbeck, 0., Finite factor coverings of groups, J. Algebra 17 (1971), 67—73. Stanley, T.E., Generalizations of the Classes of nilpotent and hypercentral groups, Math. Z. 118 (1970), 180—190. , Residual x-centrality in groups, Math. Z. 126 (1972), 1—5. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII u[I[I[[1]]I][[[]II[[][I]IH