A Thesis Submitted for the Degree of PhD at the University of Warwick
Permanent WRAP URL: http://wrap.warwick.ac.uk/140389
Copyright and reuse: This thesis is made available online and is protected by original copyright. Please scroll down to view the document itself. Please refer to the repository record for this item for information to help you to cite it. Our policy information is available from the repository home page.
For more information, please contact the WRAP Team at: [email protected]
warwick.ac.uk/lib-publications CONTRIBUTIONS TO THE THEORY
OF
SUBNORMAL AND ASCENDANT SUBGROUPS
by
Jennifer Whitehead
A dissertation submitted for the degree of
Doctor of Philosophy
in the University of Warwick
1975
•. It «I . il il A
CONTENTS
Acknowledgments (0
A b stra ct (Ü)
Introduction 1
Notation and Terminology 7
P A R T O N E
Residuals and the Lower Central Series of Subnorm al and Ascendant Subgroups
C hapter 1 The Locally Nilpotent Residual 13 of the Join of two Subnormal Subgroups
C hapter 2 The Lower Central Series of 26 the Join of two Subnormal Subgroups
C hapter 3 Residuals of the Join of two 37 Ascendant Subgroups
P A R T T W O
C rtte ria for Subnormality and Ascendancy in Soluble and Generalized Soluble Groups
C hapter 4 Subnormality and Ascendancy in 45 Soluble and Generalized Soluble G roups
C hapter 5 Subnormality in Soluble Groups 60 with Flnlteness Conditions
Appendix 78
References 82 ACKNOWLEDGEMENTS
I would like to especially thank my supervisor, D r. Stewart Stonehewer, who has been a constant source of encouragement and ideas, and who has helped me over many difficulties. Thanks are also extended to Dr. Brian Hartley, for his interest and supervision during the first six months of 1974, and whose ideas helped to shape the second part of the thesis.
I am grateful to the Science Research Council for its financial support during the period 1971-1974.
Humble thanks go to Nalsey Tinberg who has so patiently and cheerfully done the difficult job of typing.
Finally, I would like to thank Michael Mailer who has offered me invaluable support, friendship and encouragement during the months of its preparation. (it )
A b stra ct
The thesis is divided into two main parts. We begin with a general introduction and then a section on notation and terminology. In part one, consisting of the first three chapters, we are concerned with residuals of the join of two subnormal or ascendant subgroups, and the lower central series of the join of two subnormal subgroups. The results generalize those of Wielandt [35], In chapter one, we prove some elementary results on J^-residuals, for a class of groups £ , and examine, in particular the locally nilpotent residual of a locally finite group which is the join of two subnormal subgroups, and the locally nilpotent residual of the join of two subnormal subgroups satisfying the minimal condition for subnormal subgroups. In chapter two we prove results on the lower central series of the join, G, of two subnormal subgroups H and K in the cases when (l) G satisfies the minimal condition on normal subgroups and (ii) when H and K satisfy the maximal condition for subnormal subgroups. In chapter three, residuals of the Join, G, of two ascendant subgroups are studied when G is a locally finite group and when H and K satisfy the minimal condition for subnormal subgroups. In part two, we examine criteria for subnormality and ascendancy in soluble and generalized soluble groups. In chapter four we extend some of the recent work of Wielandt on criteria for subnormality in finite groups f3®]* to soluble groups and various classes of generalized soluble groups. In chapter five we consider conditions imposed on the generators of a finitely generated subgroup H of a soluble group, which imply subnormality subject to certain additional finiteness conditions. We examine first the case of a finite
eiUtMHliiMMSIf'
A b stra ct
The thesis is divided into two main parts. We begin with a general introduction and then a section on notation and terminology. In part one, consisting of the first three chapters, we are concerned with residuals of the join of two subnormal or ascendant subgroups, and the lower central series of the join of two subnormal subgroups. The results generalize those of Wielandt [35], In chapter one, we prove some elementary results on 3^-residuals, for a class of groups It , and examine, in particular the locally nilpotent residual of a locally finite group which is the join of two subnormal subgroups, and the locally nilpotent residual of the join of two subnormal subgroups satisfying the minimal condition for subnormal subgroups. In chapter two we prove results on the lower central series of the join, G, of two subnormal subgroups H and K in the cases when (i) G satisfies the minimal condition on normal subgroups and (li) when H and K satisfy the maximal condition for subnormal subgroups. In chapter three, residuals of the join, G, of two ascendant subgroups are studied when G is a locally finite group and when H and K satisfy the minimal condition for subnormal subgroups. In part two, we examine criteria for subnormality and ascendancy in soluble and generalized soluble groups. In chapter four we extend some of the recent work of Wielandt on criteria for subnormality in finite groups [36], to soluble groups and various classes of generalized soluble groups. In chapter five we consider conditions imposed on the generators of a finitely generated subgroup H of a soluble group, which imply subnormality subject to certain additional finiteness conditions. We examine first the case of a finite soluble group.
The work In this thesis, apart from the results attributed to others, is to the best of my knowledge, original. Introduction
Subnormality was first studied in 1939 by Wielandt f34]. He proves several fundamental results, here, for groups with a composition series; in particular that the join of two subnormal subgroups in this class is subnormal, and he shows how his proof can be extended to groups with the maximal condition for subgroups. The join J of two subnormal subgroups H and K need not, however, be subnormal in an arbitrary group, as an example by Zassenhaus shows ([37] Appendix D, Exercise 23). Even when H is an elementary abelian 2-group and K is of order 2, J need not be subnormal as an unpublished example by P. Hall shows. Other such examples are given in [31],
In the early 1960's Robinson [22] examined conditions imposed on a group G which ensure that joins of finitely many subnormal subgroups of G are subnormal. Since the intersection of finitely many subnormal subgroups is a subnormal subgroup, the groups for which the join property holds are just the class of groups| whose set of subnormal subgroups is a lattice (with respect to the operations set intersection and group theoretical join). Denoting this class of groups by JL in [22] Robinson shows that if G is a group whose derived subgroup G' satisfies the maximal condition for subnormal subgroups, then G belongs to i, , and that an extension of a group in the class £ by a group which satisfies the maximal condition for subgroups belongs to the class . He also examines conditions imposed on a pair of subnormal subgroups which will make their Join subnormal.
A subnormal coalition class is a class of groups, , for which a pair of subnormal 3E —subgroups generates a subnormal 36.- subgroup. These have been studied in the literature and include the class of finite, finite TT-gr'oups and polycyclic groups (Wielandt [34]), groups with the maximal condition for subgroups (Baer f3])* finitely generated nilpotent groups (Baer [2 ]), groups
i *
with maximal or minimal condition on subnormal subgroups (Robinson [23]; Roseblade [27] [29]), minimax groups (Roseblade unpublished), any subjunctive class of finitely generated groups [31], and more recently the class of groups with finite rank [5 ],
It is well-known that the join of two nilpotent subnormal subgroups need not be nilpotent, and in fact the Zassenhaus example [37] shows that this is so. Even when G is generated by two subnormal abelian subgroups of exponent 2 , th e ir join need not be nilpotent (see for example [31 ]). The fact that in all such examples as these, the join, although not nilpotent, was soluble, led to the conjecture that the join of two soluble subnormal subgroups was soluble. This had been proved by Wielandt in [34] for groups with a composition series, and in 1970 Stonehewer f32] showed it was true in general.
The nilpotency of the join J, of two nilpotent subnormal subgroups H and K, under addition hypotheses, are found in the literature, however. These include when H and K are finitely generated (Baer [2]), H and K satisfy the minimal condition for subnormal subgroups (Robinson [23] and Roseblade [27]), H and K satisfy the maximal condition for subnormal subgroups (Roseblade r29]) and H and K have finite rank [5].
Denoting the class of nilpotent groups by n. and the limit of the lower central series by G*”, in 1957 Wielandt ([35] Satz 1.5) showed more generally that if H and K are subnormal subgroups of a group G with a composition series, then
H*K - K HK and < H, K §■ = H* K* (1)
Again, denoting the class of soluble groups by A , and the limit of the derived series by G* , Wtelandt has shown ([34] Satz 2.3 and [35] Satz 2.4)
K hi* and H , K H*K* ( 2)
In both these results, the condition that G has a composition series cannot be removed, for let A be any group. Then by
A ik A ( i ti'iiM im uH M ir a
[9] lemma 13, A can be embedded in a normal product G of two free subgroups H and K. But by [13] HJl' = H^= K *1 = K = 1 and so if G^ = Gft-=1, this would imply = A^ = 1 for any group A.
Howevt -,in r32], Stonehewer shows that (2) holds provided that and are soluble. In [30] Roseblade
has generalized this still further, and shown that denoting the n-th term of the derived series of a group G by G^n\ then if J is the join of two subnormal subgroups H and K, then given any two positive integers and r^ there exists an integer r such that
(3 )
More recently, Stonehewer has shown in [33] that if H and K have finite rank, then (1) holds provided that e A- and K/j^tA .
In chapter 1'of the thesis we shall prove a result analogous to ( 1) for the locally nilpotent residual of the join of two subnormal subgroups H and K in a locally finite group, and when H and K satisfy the minimal condition for subnormal subgroups. In chapter two we shall obtain results similar to (3) for the lower central series when J satisfies the minimal condition on normal subgroups and when H and K satisfy the maximal condition for subnormal subgroups.
Closely tied to these results, is the question of permutability of subnormal subgroups which has been of interest since the 1939 paper. Wlelandt [34] showed that for groups with a composition series, a perfect subnormal subgroups permutes with every other subnormal subgroup. In [28] Roseblade extended this result and showed that if the tensor product
Is trivial and H and K are subnormal subgroups of any group G, then they permute.
Permutablltty also occurs in (1) and ( 2) since H^K, 1 *• t*
H* K, H*1" K*" , HA K* are subgroups.
In [30] Roseblade shows that there always exists some term of the derived series of H that permutes with K. However, it is not known in general whether this is true when derived series is replaced by lower central series. This is certainly true by adapting Wielandt's proof of (1) when H and K satsify the minimal condition for subnormal subgroups. In chapter two, we discuss this question for subnormal subgroups H and K of a group G satisfying the minimal condition for normal subgroups.
Finally, on the question of permutability, we note that in [17] Lennox showed that two subnormal subgroups H and K permute provided that they do so modulo each term of the derived series. Recently it has been shown by Brewster [4] that this remains true when derived series is replaced by lower central series.
In infinite groups, subgroups belonging to series of a group other than a finite series are also considered, in particular ascendant subgroups. A subgroup H of a group G is ascendant in G if there is an ordinal and subgroups
{ H s asp} of G such that H H H (a < p) a a+1 r 1 H U H If u s p is a limit ordinal and H a a < i
Ascendance was first studied by Plotkln [21 ] and Gruenberg [ 6] , and even earlier, although not explicitly, In Baer [1 ]. The Join of two ascendant subgroups need not be ascendant, as examples in the case of subnormality show that even the Join of two subnormal subgroups need not be ascendant (see for example [3 1 ]).
Examples of ascendant coalition classes, X., (the Join of two ascendant 31-subgroups is an ascendant 3£-subgroup) are the
d < lU M M liBlltSII 1 class of finitely generated nilpotent groups (Gruenberg [ 6] ) and groups with the minimal condition for subnormal subgroups (Robinson [23]).
In general, however, the problem of finding whether a class is an ascendant coalition class seems to be a more difficult problem than deciding whether it is a subnormal coalition class. It is not known whether the classes: finitely generated groups and groups with the maximal condition on subnormal or ascendant subgroups are ascendant coalition classes. Recently Hulse [12] has shown that certain sub-classes of the class of finitely generated groups are ascendant coalition c la s s e s .
Results analogous to (1) and (2) for a group G generated by two ascendant subgroups and various questions on permutabiltty of ascendant subgroups have not appeared in the literature. In chapter three of the thesis we discuss these questions when H and K are ascendant subgroups of a group G and H and K satisfy the minimal condition for subnormal subgroups.
Research in subnormality and ascendancy has in recent years been led by Wielandt to examine criteria for subnormality and ascendancy in various classes of groups.
In [36]* Wielandt has examined conditions in a finite group G which imply subnormality of a subgroup A. For example If A 1 ,... are generating sets of A such that
[ 9* * • • • ] £ A 19(6 (4 )
then A sn G.
Wielandt has also extended some of his results to groups satisfying the maximal condition, as yet unpublished.
In [10] Hartley and Peng have examined subnormality and ascendancy in groups with the minimal condition for subgroups.
I I ti i SiUilMMliiMMif!** They have shown that if (4) holds for
(t) n = n(g) and (ii) n a fixed integer then in case (i) A is ascendant in G and (ii) A is subnormal in G. Alternative proofs of results known about left-Engel elements are also obtained in both the cases when G is finite or G satisfies the minimal condition.
In chapter four we extend some of these criteria for subnormality and ascendancy to soluble and generalized soluble groups.
In [20] Peng examines subnormality in soluble-by-Max groups. We shall discuss his results more fully in chapter five, and obtatn a criteria for subnormality in soluble groups subject to certain finiteness conditions, which is weaker than those discussed previously. Notation and Terminology
Capital Roman letters are used to denote groups and small Roman letters for elements of a group. H s G, H G and H < G denote "H is a subgroup of G," "H is a proper subgroup of G," and "H is a normal subgroup of G." If X is any subset of G, then < X > denotes the subgroup of G generated by the elements of X. The normalizer of X in G is denoted by N (X) and the centralizer by C (X). C? G If V is a subset of G, we denote the normal closure of X in V by X and define it as X ^ = < x^: x ^ X, y g V > where x^ = y 'x y ,
If g1,... ,gn are elements of a group G, then
[ gr g2 ] = g , " f l ^ 1 ^ g2 and
[ 9i»....gn ] ™ [ ]* 9n ] •
If are subgroups of G, then
[ V y2 ] " < r v1 .y2 ] J yt € Yt» » = 1,2 > and
[ Y 1****»Y n ] = C C Y 1»****Yn_1 ]» Y n ] •
If V * G then Core^, V is the largest normal subgroup of G contained In V.
If p is a prime, then p’ denotes the set of primes different from p. Suppose that n is a set of primes then an element x of finite order In a group is said to be a Tr-element If the prime divisors of the order of x all lie in the set n . A group G is a yf-group If every element of G is a ^-e le m e n t.
We shalt use and assume the language of group classes and
We denote classes of groups as follows:
cf = finite groups = finite „-groups TT Oy = finitely generated groups O l = abelian groups Jl, = nilpotent groups A> = soluble groups
Max, Max-sn, Max-n = groups with the maximal condition on the set of all subgroups, subnormal subgroups, normal subgroups respectively.
Min, Min-sn, Min-n = groups with the minimal condition on the set of all subgroups, subnormal subgroups, normal subgroups respectively.
Le t 3L be a class of groups and le t G be any group. The 3E. -radical of G is the product of all the normal -subgroups of G. The ¿.-residual of G is the Intersection of all normal subgroups of G whose factor groups tn G are ¿-groups. at JE. We denote the x -residual of a group G by G . G/ G Jt is said to be a restdualty ¿-group. If G is a locally finite group we denote the „-residual, i.e. the Intersection of all normal subgroups of G whoso factors In G are L^f -groups, by On(G). TT
Series of Subgroups
Let H be a subgroup of a group G and let £ be a totally ordered set.
A series between H and__G of type £ is a set of subgroups of G [ U , V igfE} such that a o <0 each U and V contains H. 0 0 0 0 G \ H - u (tv ) V U a a A series between 1 and G is called a series in G. The subgroups U and V are the terms of the series and o o the groups U are the factors of the series. A series (i V o in G is called a normal series tf the subgroups U and V a o are all normal subgroups of G. If the totally ordered set £ is well-ordered then the series is said to be an ascending series. In this case we obtain (see Robinson [25] p. 11) H » V « V ^ ...... V * V ...... V = G O 1 rr n* • n where V ■ n V If \ Is a limit ordinal ' »«X 8 If e ' denotes £ with the reverse order < i.e. n < i if and only tf T < g , then tf j' is well-ordered the series is a descending series. A subgroup H of a group G is termed serial in G If there is a series between H and G. Should the series be ascending, H is said to bo ascendant in G, and tf the series has finite length, H la said to be subnormal In G. We w rite H ser G, H asc G, H sn G . If there is a series between H and G with order type £, wo write H G . liiBununttNian*' Let H £ G and let a descending series of subgroups of Let H & G and let i be a non-negative integer; then H = H y*G H and H G if and only if y*G H s H (see Robinson [22 ]) . We denote the n-th terms of the derived series, the lower central series and the upper central series by: Yn We denote the derived subgroup of G by G'. If G = G', G is said to be perfect. ^(G) = ^(G ) denotes the centre of G. The limit of the upper central series of a group G is called the hypercentre of G. G is said to be a hypercentral group if G coincides with its hypercentre. G is called a Gruenberg group if G can be generated by its ascendant abelian subgroups. We shall say a group G has finite (Prufer) rank, r, if every finitely generated subgroup can be generated by r elements and r is the teast positive integer with this property An abelian group has finite rank r if and only if r «= r + max r <• m ° P P whore the p-component of A is a direct product of r P cyclic or quast-cyctic groups A 1 , and the factor group of A with respect to its torsion-subgroup is isomorphic with a subgroup of a d ire ct .Si/M of r but no few er copies of the o additive group of rational numbers. GL(n,F) ■ General linear group of alt invertible nxn matrices over a field F. Other, more specific definitions are given in the text as they are required. Mill * All RESIDUALS AND THE LOWER CENTRAL SERIES OF SUBNORMAL AND ASCENDANT SUBGROUPS I Chapter 1 The Locally Nil potent Residua! of the Join of two Subnormal Subgroups. Sec. 1 Elementary Results In this section we prove some elementary general results that wo will use later in the thesis. We begin by examining the effect of homomoi'phisms on the 3£ - residual of a group G. Lemma 1 .1 Let it = Q i and suppose that G is a group such that G ^x 6 36 • Then if e is any homomorphism of j. * G , we have (G 0 ) = ( G ) 0 and GJ/( g g)*1 e ^ * Proof Let N = Ker 9. T h e n N 4 G and ^ < * 1 y N* < = > * - * • Hence ( G 9 ) s ( G * ) 9 . Now let N s L s G be such that L ^ «j G ^ N and •* I* Therefore L < G and G ^ *. £ _3fc • Hence G* s L and G* * L^ . Since this is true for all such L ^ we have (G*) 9 i (G 9 )* and hence equality. AUo °*<6o? ’ °>*1Lo Therefore G Q f 36 . We next examine the 36 - residual of the Join of a subnormal and a normal subgroup. Lemma 1 .2 Let 3i = N 3E. = S £ = Q 3: . Suppose that ------o n G = H K where H sn G, K ^ G and that at € i > 1 * « « * • ^ * 3E Th e n G ^ * 6 £ and G 1 = H*K* Proof Let N G such that G ^.. T h e n ^ N € 3t • € S n z = x . Since E. we have !^i< n n - K > * « 3 z z VI X and so K ^ ^ K q • n N € 3r X Since this is true for any such N we have K s G . Similarly, H ^ s G * . I % i f Now K char K^G K G . Since K s G* , » t using lemma 1.1 we may factor G by K and assume K = 1 £ * £ K Hence we must show that H = G By above, H s G Let H . K > where HjxpX € 3E. «•* K H^h» ^ H* € Q* = * . So G^J'x Is generated by two normal 3£ - subgroups and by N - closure we have G ^ x € 3t • i at Hence G * s H** and we have equality and G ^j^ € 3L • Now suppose m > 1 . Let H^ = H , Then H , - H 1 n H K = Hi ( Ht n K ) where n K € Sn £ = Z . m -1 Since H H1 by induction H. H 3^ and V « ,* i3fe • X 3t Then, by the case m ■ 1 , wo have H^ ■ G and G/ g * € *• I t Hence H = H = G and the lemma is proved. Although G/^X is always a residually — I group, it is not necessarily an - group unless G satisfies the minimal condition on normal subgroups, or unless the class 36 is R - closed. In the case of locally finite groups, however, we have the following known result. Lemma 1 .3 Let 3t = < S, R q > s if". T h e n RLlnL^ = Li. U t «L3tf>«-3F Proof Obviously, L £ R L 36. p L Let F be a finitely generated sugbroup of G. Since G ( L?- then F g ^ . Let 1 x € F . Since G g R L jfc there exists a x f K (x ) . Now F/p n K (x) * F K (x ^ (x) which is a finite subgroup of the L 1 - group G/K . ^ Hence is contained in some 36 - group. Therefore, F K (*)/£ ^ € S 36 = 36 . Now n ( F n k (x) ) = ( 1 ) 1*x€F S o , f £ r 3L = 3b . o Hence G g L 36 , The following result relates the L St - residual of a locally finite group G to the 3L - residuals of its finite subgroup Lem m a 1 .4 Let G be a locally finite group. Let f be a class of groups such that 9E = < S, R q > s ¿f- . Th e n if 9t is also Q — closed we have L3t 3t G = < F | F is a finite subgroup of G > . Proof By lemma 1 .3 we have that G/£.LX g l 3E . Le t R = < F j F is a finite subgroup of G > . Then R F Ry^ — p R which is a homomorphic image of FyjLjj. £ R0 IE = ^ • Therefore F Ryj^ g Q 35. = X. . Now let Ky^ be any finite subgroup of * Let K/ r = < R k1 , • • •, R > • T h e n = < • *^n £ 3t by the above. l3L „ Hence G y ^ £ Lit and it follows that G s R . However, if F is any finite subgroup of G, FG/ S L* t SX . X. a n d T h e re fo re , F ^ - s F p G^- ^ s G^" ^ , So R s G*~^ and we have equality. Sec. 2 Permutabtllty of Locally Nilpotent Residuals Wlelandt has shown in £ 5], that for a finite group G, generated by two subnormal subgroups H and K G *1 = H>VKn' and H^K = K H1'”. We are concerned here with similar results for the locally nilpotent residual of a group G, generated by two subnormal subgroups, and shall show that permutablllty occurs when L i t L i t H K > and H/j_|LJt g LIT, As Stonehewer points out in [33]» the difficulty in extending the Wielandt proof of permutability of nilpotent residuals is that his proof depends on the fact that for some integer r, defining H** ^ (H^y -h1-1 * h A = ( h?1 ) we have H.it is a perfect subnormal subgroup, and hence permutes with all subnormal subgroups of G (see Wielandt ). In general the series H . * ^ • will not terminate, and so we do not know whether for some integer r, is perfect. However, following the method in [33] we use the following generalization by Brewster in w of a re su lt by Lennox [1^ ] . Lemma 1.5 (Brewster) Suppose that H and K are subnormal subgroups of a group G, that G = < H, K > and that for all finite c a 1 , G = H K V (G). Then G = H K. A proof of this result is included in the Appendix. We are now able to generalize theorem A of [jag and obtain: T h e o re m A L e t G = < H , K > where H , K are subnorm al subgroups of G, and suppose that L it L3X = < M 1 ^ > • The n G*-^ = HL*''KLi'‘ provided that € l K. P ro o f Let M = G*- ^ - and for some Integer c s 1 let N = Yc (M ) . Hence M < G . Since the class L lT - satisfies the hypothesis of lemma 1.2 and since by lemma 1.1 I V'fiuwiHiiaiiHffir ( H Ljv C LJV , we may apply lemma 1 .2 / X H > ^ N ) to the product H = (H ^N )( M/ N ) • l Jl Now M/ n g ft and so (M ^ ) = 1 Hence ( H ) L,V = ( H ) LJl" By lemma 1 .1 we have (H ^ = and therefore ^ H ) = H N^j anc* hL31X N 4 Hf^N * So N have M = H*-* K1-^ Yc(M ) . As this is true for all finite c ^ 1 , by lemma 1.5 we have M = H L Jt Corollary Let G = < H, K > where H, K sn G and I suppose that , L f t L i t K L i L - < H Th e n If hJ^LTL € Lh, , we have L i t G,J'- - H1^ and H ^ K K H W K ^ = Proof This follows from theorem A and the next lemma. Lem m a 1 .6 L e t G = < H, K > and let X be a class of groups such that G* = H* K* . Th e n H * K = K and H = H K • * Proof K H* = K K£ H* = KGJ = G* K since G Sec. 3 T h e Join P roblem In this section we shall be concerned solely in proving that for a locally finite group G, generated by two subnormal subgroups H and K we have G although it generalizes the methods used by Wielandt [35]> is independent of the finite case. In chapter 3, we shall discuss the locally nilpotent and locally soluble residuals of a locally finite group generated by two serial subgroups, and give an alternative proof which reduces to the finite case and the Wielandt theorem We shall need the following well-known structure theorem for locally nilpotent, locally finite groups. Lemma 1 .7 Let G 5 L( J fl JX)* Then the set G^ of p—elements of G is a subgroup of G, and G is the direct product of the G . P Proof See Kurosh [|(,] Vol II p. 230. We shall also need L e m m a 1.8(Htrsch - Plotkln - Baer) If 3E. - < N » S > 36 s M ax then L 36. is N' - closed. Proof See Robinson [jj^] p. 57 Theorem 2.31 . We begin by examining the n — residual O^CG) of a locally finite group G. Since the class Is S and Rq - closed we have by lemma 1.3 that G ^ tvG) is a locally finite n - group. kl\J Lemma 1 .9 Let X, V be subgroups of a locally finite group G such that G = < X, Y > and Y sn G. Suppose that every finite subgroup of X lies in a subnormal subgroup of G in X. If On(Y) s X n Y then O^G) s X . Proof Let Y G and proceed by induction on m. Case 1 m = 1. Then Y <, G , By lemma 1 .4 O^G) = < On(F) | F is a finite subgroup of G > (*) Now if F is any finite subgroup of G, then F is contained in some finite subgroup F 1 where Fr1 = < x1,...,x n,y1,...,y m > where xt € X, y^ e Y . Therefore F1 = < X p F , Y p F^ > . By hypothesis there exists a subnormal subgroup X of G such that x n F1 S X1 s X . Hence F1 = < n F1, Y n F1 > and X, n F, sn F^ Y n F, 4 F, . Since F is finite X p F ^ 1 1 X t n F 1 ) * " and Y n F. v» e *£. and since the class ^ >6"CY n F, ) ' satisfies the hypotheses of lemma 1.2 we have by lemma 1.2 O^CF,) - o"( X 1 n F 1 ) o"( Y n f 1 ) s x . Therefore O^CF) s X for all finite subgroups F of G and so by (*) On(G) s X as required. Case 2 m > 1 Suppose by Induction that the lemma is true when Y has subnormal defect less than m . Let Y^ = Y . Now If F Is any finite subgroup of V f) X, then by hypothesis there exists a subnormal subgroup S of G such that F s S s X . Then F £ S p Y^ £ Y^ p X and since Y^ S n < S sn G , i.e. S p Y sn G. Hence the group Y = < Y^ p X, Y > satisfies the m™1 *. ___ hypotheses of the lemma and Y Y since Y £ Y . Therefore, by induction we have O^Y) £ X p Y^ . Let M = O^CY), X = X p Y and N = On(*) . Then M s X . Using lemma 1.3 we have > 6 < L 9 -„ a" d « l 5 -„ • Since O^CM) char M char V we have On(M) X . Therefore, by the P — closure of the class x/o ”(M) * L 3 -n • Hence N = O^CM) char V and so N = O^Y) and N <] G. S ince * Y % € I-3- y N « TT ' IN TT • 0 £ £ > - n Now -- - y G n/ n | x e X > . - < n < -< S ince 'If € L- 2 ~ x * TT we know that Y . is generated by subnormal - subgroup TT X n Since the class satisfies the hypotheses of le m m a 1 . 8, TT |_^i- is N*- closed and hence Y y g L jf- . TT X n v We have reduced to case 1, applied to the group g/ n = < x/ n , y i / n > slnce y i / n < g/ n * Hence we may deduce that ° " < g/ n ) ‘ x / n • By lemma 1.1 On (G/N ) = ° nCG >N /N * Therefore O^G) N 4 X and O^G) s X . Lem m a 1.10 Le t G be a locally finite group. Let A, B be subnormal subgroups of G and let J = < A , B > . Then if F ts any finite subgroup of J, there exists a subnormal subgroup S of G such that F < S s J . Proof This follows immediately from Roseblade and Stonehewer [jjj] Theorem A. It is now an easy consequence of lemmas 1.9 and 1.10 to prove Theorem 1 .1 Let G be a locally finite group generated by two subnormal subgroups H and K. Then On(G) = < O^H), On(K) > . Proof Since ^ oTT(G) a H 0"(G^qTT(G) € L ^ we have O^H) s On(G) . Similarly On(K) s O^CG) , and so < Q"(H), On(K) > s 0"(G) . B y le m m a 1.10, the group < O ^H ), On(K) > satisfies the hypotheses of the subgroup X in lemma 1.9 . Hence in lemma 1 .9 let G = < On(H), K > , X = < O^CH), O^CK) > and V = K. Then we have O" ( < Q"(H), K > ) s < 0" Agatn in lemma 1 .9 let G = G, X = < O^H), K > and V = H. The subgroup X satisfies the hypotheses of lemma 1.9 by lem m a 1.10 . Hence we have On(G) s < O^H), K > . Let L = O™ ( < On(H), K > ) . Then L s < O^H), On(K) > and L n Since O^CG) char G, we have by lemma 1.3 that So O^G) s L < < O^CH), On(K) > and hence 0"(G) = < 0"(H), 0"(K) > . We now prove the main theorem of this section. T h e o re m B Let G be a locally finite group generated by two subnormal subgroups H and K. Th e n G/gLJX , H/HL)X , Ky^LlX c LjX LJX l jx LJX. and G - < H , K Proof By lemma 1 .3 we know that Gy^LlX , Hy'^Lj'L , Ky'^LJl- are locally nilpotent. By theorem A we know since Hy' |_|L_JV £ LlX that H*~^" and permute, provided that their join is equal to L)L U t, . Hence it is enough to prove that G ^ H , K Now H y 'n n G U V = H G ^ g U V f U V and therefore H L j l , . Similarly G*-^ and so LJX LJX LJX < H , K > S G In theorem 1 .1 let n = { p } and apply to the group G generated by H and K. Then 0 P(G ) = < 0 P( H ) , 0 P(K ) > . B y lem m a 1.7, is the direct product of p-groups. Hence 0P’(0P(K) ) s . Let L = 0P(G). Applying theorem 1.1 to the group L w ith tt “ { P' } we have Op'(L) = < 0 P’cOP(H) ), oP'(oP(K) ) > . LJX P Now G s O (G) and so by lemma 1.3 ,LJX ,LÌL is an , - group. / CoreG< H*-*1, , K This Is true for all primes p. Hence _LjX LJV LIT H . K 25 Corollary Let H, K g Min—sn and let H, K be subnormal in their join G = < H, K > Then G/^LJV , HlyS^Lth are locally nilpotent and o'-*- - * - Ha’-KLJU . Proof By the subnormal coalescence of the class Min-sn (see for example gjj]) we know that G g Min-sn . T h e re fo re G ^ / q LJI- , K/^LJl, , are locally nilpotent, since H, K and G satisfy Min-sn. Now H/ h LJ1. , K / KLiV are locally nilpotent groups satisfying Min-sn and therefore they satisfy Min and are soluble (see Robinson [^y] p. 154 ). By the condition Min-sn we have G^<& » » anci Ky^i, are soluble and so by Stonehewer [y] we have H u i . K i.n . Hence, without loss of generality, we may assume G = 1 . and G is soluble. Since G satisfies Min-sn, G is ¿Sernikov and locally finite. LJV u x LJL ( 1Lh . L J L Hence G = < H K H K by theorem B . C h a p te r 2. The Lower Central Series of the Join of two Subnormal Subgroups. Sec. 1 The Minimal Condition on Normal Subgroups. We consider the nilpotent residual of a group G, generated by two subnormal subgroups H and K, which has the minimal condition on normal subgroups. The condition Min-n of course ensures that the nilpotent residual of G has nilpotent factor group. However, since the condition Min-n is not necessarily inherited by normal subgroups of G (See Robinson p. 153 ), we have no guarantee that and K^Jvare nilpotent. However, we are able to prove more generally: Theorem C Let G g Min-n and let H and K be subnormal subgroups of G such that G = < H, K >. Then for any positive integers , r 2 we have that s Y (H ) Y (K ) r t r x For the proof we shall need the following result on the derived series of the Join of two subnormal subgroups, given ‘ n bo]* Lemma 2.1 (Roseblade) Let H and K be subnormal subgroups of a group G = < H, K > and let a» 3 be ®hy positive integers. Then there exists an integer j such that g ( 6) s h ( ol) K ( p ) We shall also require the following well-known results on groups satisfying M ln-n. Lemma 2.2 (Baer) A soluble group which satisfies Min-n is locally finite. Proof See Robinson [Is’j p. 153. Lemma 2.3 (Carin, Me Lain) The locally nilpotent groups which satisfy Min-n are precisely the hypercentral Cernikov groups. Thus for locally nilpotent groups, the properties Min and Min-n co in cid e . P ro o f See Robinson [a$] p . 154. Lemma 2.4 (Baer) G is a nilpotent group satisfying Min if and only if ^(G) satisfies Min and ls a hilpotent group. Proof See Robinson [a«g p. 69. Wielandt has shown in [36J» for groups G with a composition series, and generated by two subnormal subgroups H and K, that once the Join property G*1, = < H^"f K^*> is established for groups in this class, then we have permutabllity of the residuals, t.e. G**" = Hn-K*\ Wtelandt's argument can be adapted to groups in the class Min-sn, and this we prove In lemma 2.5. Following Wielandt, we define G* inductively by: „ r «.r - 1 G*1 = ( G*1- ) ftnJ Suknerm J Lemma 2.5 Suppose that for a group G »¿satisfying the minimal condition on subnormal subgroups and generated by two subnormal subgroups H and K, that G^1, = < H* «*•>. Then JH**- permutes with and so G^*“ Proof Since H is subnormal In G, H satisfies the minimal condition on subnormal subgroups. i i i M K n a n w r Hence for r sufficiently large and is a perfect subnormal subgroup of G. By £ 8] * r we obtain that Hrt permutes with all subnormal subgroups of G. In particular is a subgroup. Now suppose that for any integer l, that KnHtt is a subgroup. Then i v»1“ 1 :*H, t = K h Hh' H 11* * 1-1 Now H*1, , K g Min-sn and since Min-sn is a subnormal coalition class [¿3], the group < K, H*'- > £ Min-sn, and is subnormal in G. By hypothesis < K, H**" H**' >. j, . i - 1 * 1-1 * 1-1 H ence, K 1 l-T1 = < K, H > H and since * 1-1 h ni_1 < K , H11 > is normalized by H we have that K ^ H * 1“ 1 is a subgroup. By induction on l decreasing we obtain K ^ ’ H*'" is a subgroup. We are now able to prove theorem C. Rr-pof of theorem C. We know that there exist positive integers a and p such that H (o) s Y So by lemma 2.1 there exists an Integer j such that — <6) s Y Y b r \ rx Now y . ( V * and similarly v r < h g ^ X g (6)> = rr ^ G y G^ 2 2 Hence, without loss of generality, we may assume G ^ and that G is soluble. By lemma 2.2, G is locally finite. By theorem B we know that the factors G/gLlt, H/^LM, V k 1* are locally nilpotent and that GUV = HUlKLn s Y rCH)Yr,0<) 1 2 As before, we may assume G^" = 1 so that G is locally nilpotent. But by lemma 2.3 G satisfies the minimal condition. Hence H^n, and are nilpotent and * y V 1-1) Yr2 We wish to show that G**’ = < H**’ , K^> for then by lemma 2.5 we know that G**" = H^K***. Let X = < H*1-, K1V> . Then X G** sin ce H ^ (Jn. - H € f t => H *1 s Gn , and similarly K ^s G11- . Now by lemma 2.3, G is a ¿ernlkov group, t.e. ar. extension of an abelian group satisfying Min by a finite group. Let F be the finite residual of G. Then F is abelian and G ^ is a finite nilpotent group. Now by lemma 1.2 ( H F = H^’F*'" = H*1 since F is abelian. Similarly ( K F = K*'". Hence without loss of generality we may assume that F s H p K. Also since G,/'. n*'P°tent we bave G ^ s F . Consider H^^H. The finite residual of H^tv is • Since » / » * ■ is nilpotent, we have by lemma 2.4 that < Jry^Ti. S C C hx/'H*V > * S ° e v e ry sub9rotJP l/ h »T- contained in lySyfL Is centralized by H, and so H normalizes every subgroup L such that s L s F. In particular, H normalizes X. Similarly K normalizes X. Hence X Now G/x = < H/x , > Since g ft. , is generated by two subnormal nilpotent subgroups and G ^, satisfies Min. Hence by gjj] we have that G/x £ ft end so X. Hence G^ = < H^- , > and the theorem follows since = H ^ K ^ i Yr 1 CH > Yr 2^K ^ We Introduce some notation before proving a corollary to the theorem. If H and K are subgroups of a group G, then the permutizer, PH(K), of K in H is defined in [Sl] to be the largest subgroup of H which permutes with K. Corollary Let G g Min-n and let G = < H, K > where H and K are subnormal subgroups of G. Then there exists a positive integer j such that Y (H) «; 6 H Proof We have that G^" s H K from theorem C. Hence if L = G* K, then L = ( H p L ) K, and since L is a subgroup we have that H f) L permutes with K. Hence H p L * P (K) . Since Y (H) £ P H for sorhe 6 H 6 we have that Y (H) s H p L. 6 Hence Y (H) PJK) . 6 M We note that the proof of the corollary is taken from that given in the case of the derived series in (see (So) corollary to Theorem B). Sec, g The Maximal Condition on Subnormal Subgroups. We begin with an important theorem of Mal'cev which relates subgroups of a polycyclic group to the subgroups of finite index that contain them. Although the Mal'cev theorem is true for a slightly larger class of groups than polycyclic groups, we shall only be concerned here with the polycyclic case. We shall need this theorem several times tn the thesis and since it is not readily accessible in the literature we give a proof here based on that given by Mal'cev in [|j] . Theorem 2.1 (Mal'cev) Let G be a polycyclic group. Then any subgroup of G is equal to the intersection of all the subgroups of finite index that contain it. tl«BHMMMFlitSIIMISr Proof We use induction on d, the derived length of G. If d = 1 , then G is abelian. Hence every subgroup of G is a normal subgroup of G, and since each factor group is residually finite, the theorem follows. So assum e d > 1 arid le t G 1 be the derived group of G. Let H be any subgroup of G and let F be the intersection of all subgroups of finite index containing it. Then obviously H £ F. Suppose for a contradiction that H < F. Hence we can find an element f g F such that f j H. We consider f j H G„ . Since f is contained in the intersection of T subgroups of finite index containing H G^ , we may , without loss of generality, factor G by G^ and assume G^ = 1 and G is abelian. But then by the case d = 1 , we obtain f g H, a contradiction. C a se 2. f g H G r Then f = h^ Le t 91 h1 € H . £>! e • T = H n g 1 . Th e n ^ T and has derived series of 9 1 G 1 length d - 1 . Hence by induction there exists a subgroup F^ of finite index in G , containing T such that » . 1 F r Le t s . Th e n I G 1 8 F 1 I = s * G < G , SO and • G^ char 1 G1S" Suppose that f g H G ^ - . Then f = h2 g£ where g H, 01 s CM g G 1 • , and h2 gg = h , 0l • So h 1 _1 h2 = 91 92_1 g H n G , = T - So 9l g H G , * 1' w hich is a contradiction to ( 1 ). So f j! h G l sJ. Hence, without loss of generality, we may factor G si b y G 1 " and assume G , € * and G ^ S‘ = 1 . ■U rliUllMMIiiHMiL So | H G 1 s H | = | G 1 : H n G 1 J = t, say. t> - i t* Let M = H G 1 . Then M ‘ s H . So f i M • and we may t' 1 ' assume M ’ = 1 and M is finite. Then H is finite, of order n, say. Since G is residually finite, for each h. g H i = 1 ,....,n there exists a normal subgroup N. of G such that | G : | < „ and N. . f N t ■* ni I L e t N = 0 N. . Th e n | G : N | is finite and 1=1 1 H N . But H s H N and | G : H N | is finite, a contradiction. W e also use the following lemma. Lernm a 2 .6 Let G = H N where N y J G ) s Yr CH) Yr (N) . r r 1 1 Proof Since N H = H < H 4 . . . < H . Now for each i, H J_1 L e t l = n-1. Then by Fitting's theorem we obtain that H is nilpotent. Hence there exists an integer c^ 1 such that Yc uiM M KriaiKUi! ▼ a i. i * vs *voo* Suppose that we have found an integer cj_ 1 such that Y c * Yr C H ). 't-1 Then by Fitting's theorem H > is nilpotent. ' VYc Yc. By induction on i decreasing we obtain an integer c q = r such that y^(G) s (H) and the lemma is proved. We shall now prove the following theorem on the join of two subnormal subgroups which satisfy the maximal condition on subnormal subgroups. Theorem D, Suppose that H and K are subnormal subgroups *1 of their join G = < H, K > , and that H and K satisfy the maximal condition for subnormal subgroups. Then, given any two positive integers r^ and r # there exists a positive integer r such that YrCG) * < Y r Yr 0<) > * 1 2 Proof By the subnormal coalescence of the class Max-sn, , we have that G £ Max-sn. Now there exist positive Integers s„ , s such that 1 2 ► W * Yr Then by lemma 2.1 there exists a positive integer s such that G (S) £ H ^ l 5 K(S 25 s < y (H ), Y V '{¡WMIIfliiHIHiKli:1 G is soluble. Since G £ Max-sn, G is polycyclic. Let G have derived length d. We use induction on d. If d = 1, then G is abelian and the theorem is trivially true. So assume d > 1 and that the theorem is true for soluble groups of derived length less than d. Let N be the last but one term of the derived series. Then G/j^ has derived length d-1 , and by induction there exists a positive integer r^ such that Y (G) * NX. r 3 By lemma 2.6 there exist positive integers t^ , t^ such that Vt(NH) * Yr So replacing ^ , r 2 by t<, tg respectively, we may assume that N £ H p K . Hence N normalizes X and so X <, X N . Now X ty x ~ % n x € ei Let L = Y (G). 3 Th e n L ' < ( X N ) ' s X, and since L' Let M = X L. Then X < M and M/x a- L/\_n x * Ot Since M is polycyclic, there exists F Then Mn s F, so we have | M ! MnX | is finite and MnX ^ is torsion-free. Now by theorem 2.1, there exists a subgroup F q of G such that | G ! F q | < « and M ° X = F p M . o " By Wlelandt [35] we have that there exists an integer r 4 such that Yr (G) * * CoreG(Fo) ‘ 4 Let R = Core_(F ) and let r _ = max } , then o o 5 i V r 4 Y r (G ) 5 X R n M 5 = X ( R n M ) ) = X M n , * X ( Fo n m and that Hence we may suppose that L = V r f ) o is torsion-free. By the same method we can now show that for all primes there exist positive integers r(p) such that vKp)tG) s xmP ■ Let l be the rank of G. , l Then ,p has order less than or equal p . / X M Since L = v (G) we have r 5 v ..(G) < X Mp for all primes p. ' r j H D Hence Y r +i^G^ « fl x • 5 p Now ( ) P = X and since is torsion-free abelian of finite rank n ( ^ ) P " * • Hence n x m p = X and Y ..(G) = * • T r +1 p 5 Let r = r_ + l . Then o Y_CG) s < y Sec. 1 Locally Finite Groups. We begin with an alternative proof to theorem B, which also works for serial subgroups. The proof, however, reduces to the finite case and the Wielandt theorem given in [55] . Theorem 3.1 Let G = < A, B > be a locally finite group, where A and B are serial subgroups of G. Let & = Proof By lemma 1 .J we have G/^U (■ g L*< . Let N = GL3t and M = < AL* , BLX-> . We will show first that M = N. Now a/ a n N * Ar% e L ( * n3t) So A^-^- s A n N s N. Similarly B*- s N. S o M N. By lemma 1.4 N = < F* | F is a finite subgroup of G > * So it is enough to prove that F s M for finite subgroups F of G . Let F be any finite subgroup of G. Then F is contained in some subgroup F^ where F^ = Then F is finite and f 1 = < F-, fl A* n B > • OD S o F ' X < f , n A, F1 n B Since A is a serial subgroup of G, A p is subnormal in the finite subgroup F . Similarly B n F 1 is subnormal in F . So, by hypothesis f *■ s < A n Fi * B n Fi >* = < (A n Fi )* » ( B n ) > s M So N s M and we have a f G - < 1-* ’ > • L i „ L X , „ a L X To show permutabillty of A and B let a £ A and b f. B*- ^ - . Then a b g G*- *" and as before . L i < F * | F is a finite subgroup of G > . So a b c E* . . . E * where E is finite 1 s i s s. ^ 1 s > UJ A • • • • Let E = < E 1 • T h e n a b < E *■ and E is some finite subgroup E 9 such that E n A , E n B > o ~o ~ < “o " "• “o i — Hence a b £ E < E * o - < A n E 0 * B i , , B ( A n E0 ) Ci i , ■ ( b n E 0 ) C - i BL l A L>t L * , L * e a LX Hence a b f B A So B A Sec ■ 2 Groups with the Minimal Condition on Subnormal S u b g ro u p s■ Let G £ Min—sn. Let F (G) be the smallest subgroup of finite index of the group G. Let E (G) = F (G)*. Then E (G) is the smallest subgroup with Cernikov factor group. The following lemma is proved by Hartley and Peng in [(c>] . We include their proof as it has not yet appeared in the literature. Lemma 3.1 Let H, K g Min-sn and suppose that H and K are ascendant subgroups of a group G. Then E (H) < N(K) . Proof Suppose for a contradiction that the result is false so that there exist ascendant subgroups H , K q of a group G which satisfy Min-sn but are such that H_ = E (H ) o 1 o does not normalize K . Among the subnormal subgroups of o K which are not normalized by H , let K be minimal, o 1 i and le t G 1 = < H 1, K 1 > . By construction, normalizes every proper subgroup of , and hence normalizes also their product L. Therefore L < , and clearly L . Letting G = , H = H, l/ l , K = ^ L/ l we now have, using the fact that for any group G, E (E(G) ) = E (G) (since the class of Cernikov groups is extension closed) H, K asc G ( 1 ) H = E (H ) ( 2 ) K is sim ple (3 ) H (4 ) i n g (K) From Robinson ([¿ 5 ] Lemma 4.3 and Corollary) , we obtain H «j2 G (5 ) 2 If K = K', then it follows from (1) and (3) that K G . This follows by a simple Induction on the length of a series connecting K to G , using the well-known fact that a non—abelian simple subnormal subgroup of a group is at most 2 - step subnormal. But then H s N_(K) since F (G) normalizes every subnormal subgroup (see [aYj ) . Hence (4 ) is contradicted. Therefore K is cyclic of prime order p, and the Q normal closure K = K is a locally finite p - group, by ( 1 ) and [as*] p . 20 according to which the class of locally finite p - groups is closed under forming joins of ascendant — — G subgroups. Now from (5) we have [K,H]^KpH — R and [ R, H ] £ K n H = s*a normal locally finite p - subgroup of H. Since H satisfies Min-sn , so does S, and hence S is Bernikov ( fa$] p. 171 ). Now is finite and F (S) is the union of its finite characteristic subgroups; since S H' = H centralizes S. Hence H stabilizes by conjugation the series 1 of K and we find successively that H' = H centralizes R and then K . We now have H s NG(K), a final contradiction to (4). We examine E (G) in the following case: Lemma 3.2 Let G = < H, K > where H, K asc G and H, K g Mln-sn. Then G f Mln-sn and E (G ) = E(H)E(K) . Proof The class Mln-sn forms a subnormal and an ascendant coalition class (see fAJfl theorems 4.1 and 4.2 ). Hence G £ Min-sn. Now E (H) H and so E (H) e Min-sn; similarly E (K) g Mln-sn. Since E (H) and E (K) have no proper subgroups of finite index, they are subnormal in G, and normalize each other ( [a'i] lernma 4.3 ). Hence from above E (H) E (K) is a subnormal subgroup of G satisfying Min-sn. Let X = E (H) E (K) . Th e n E (H ) E (G ) and E (K ) s E (G) . Hence £ 2 X s E (G ), and X « E ( G ) , by lem m a 3.1. Hence X We consider L = < H , E(K ) > . Then by lemma 3.1 E (G ) S n g (H) and hence X s N q (H) . S o L -• X H . I Now X Then X H^H = XH H/^, H - H^H ^ H which is Cerntkov. .H A ls o X xH n x H «= x ( x n h ) H H which is Cemikov. and therefore X /^ e? X p H/£ ^ h Since | X H : N x H(X) | < „ (see frt] or C»»] ) and by the P-closure of the class of Cernikov groups ( £ 5-] p.69 ) X H is Cernikov. >£°rex H(X) Hence E ( X H ) * X . But E (H) £ E (X H) and E (K) = E ( E (K) ) £ E (X H) . Therefore E (X H) = X and X is normalized by H. Similarly X is normalized by K. So X G/X = < H , K X^, > is generated by two ascendant Bernikov groups, and so is Sernikov by [jj,^ theorem 4.2. Hence E (G) £ X and so E (G) = E (H) E (K) . We are now able to prove: Theorem E Let G = < H, K > where H, Kg Mln-sn and H, K asc G. Then G^ a e * and G * = < H -6 , K A > = H*KA. Proof Since Mln-sn is an ascendant coalition class we have G £ Min-sn. Hence A £ Jb • Let A = GA and B = < H Since, as before H^ £ GA and K -5 s G . VI < m Now G^. G) is Cemtkov, and hence is Now ( G ^ e (G)> “ ( (G ) and by the Q-closure of the class we have by lem m a 1 .1 c °sk (G) g * e S in ce is locally finite, we may apply theorem 3.1 (G) to obtain A E (G) = B E (G) = H6 K * E ( G ) . Now G,£ is Cemikov. Hence E (G) ^ A. Similarly, E (H) s H4 , E (K ) KC* . By lemma 3.2 we have E(G ) = E (H) E (K) . Hence G* = hf K^. Corollary Let H be a perfect ascendant subgroup of a group G, and let K be an ascendant subgroup of G. Then if H, K g Min-sn H permutes with K. Proof This follows from theorem E and lemma 1.6 . Theorem F Let G : = < H 9 K > H, K £ Min-sn and H, K asc G. T h e n G/Q-tl g L3V and GL?t = < h l t v KLn-> - HlJVK L ’ 1 Proof As in theorem E, we obtain that G g Min-sn. Hence G ^ L J L g L it . Locally nilpotent groups satisfying Mln-sn are Cernikov and soluble (see, for example, [¿q p. 154 ). Therefore H/^UV » , and G^LJl are soluble. By theorem E „A H * K * . So without toss of generality, we may assume G^ = 1 , and is soluble. But then G is ternikov and so locally finite. I I li («UHMHHiHHSIL Hence by theorem 3.1 GLJ>- - H ^K 1^ . We note that the equivalent theorem is not true for nilpotent residuals, since the join of two ascendant nilpotent subgroups which s a tis fy M in -s n need not be nilpotent. 0 II Let G = < t, A;afc=a 1, ~I 1 . a g A > where A o. Cgo, . Then T = < t > and A a re ascendant in G and belong to Min-sn q JV . But G is not nilpotent. P A R T T W O CRITERIA FOR SUBNORMALITY AND ASCENDANCY IN SOLUBLE AND GENERALIZED SOLUBLE GROUPS ■ 4.1 Chapter 4 Subnormality and Ascendancy in Soluble and Generalized Soluble Groups. The results in this chapter are based on those proved by Wielandt for finite groups in [34,]. Following Kurosh [|fc] we use the following definitions: A class of generalized soluble groups is a class satisfying ? n l s A « ^ i.e. a finite 3t-group is soluble and every soluble group is an group. A group is SN* or SI* if tt has respectively an ascending or ascending normal series with abelian factors. S e c . 1 Soluble Groups. The main theorems in this section will be: Theorem G.1 Let G be a soluble group of derived length d, and let H be a subgroup of G . Suppose that for all sequences h1 ,.... .h^; hj ( H, n a fixed integer, and fo r a ll g g G [ 9* ^ > • • • • nd Then H < G . Corollary G.1 Let G be a soluble group of derived length d and H a subgroup of G . Then tf H n < H,g > vg € G» n a fixed integer, then H 0(1 G . The corollary follows easily from the theorem since if L = < H,g >, some g € G , by hypothesis we have Y L h "* H . thiiuM iiiH atns; I Hence the subgroup H satisfies also the hypothesis of the theorem. Theorem H.1 Let G be a soluble group and let H be a subgroup of G . Then H is subnormal in G if and only if there exists an integer n * 0 and for each g g G a sequence H. H of generating sets of H such that T g,H H ] s l 1 n '■in For our proof of theorem G.1 we examine closely the following special case: Lemma 4 ■ 1 Let G be a group and let H be a subgroup of G , and suppose that G = HA where A If, for all sequences h ...... h^; h^ f H, n a fixed integer, and for all g £ G [ 9 , ------,h n ] 6 H then H «q1 G . Proof Since A < G we have A f| H Now let a g A. Let g1 ...... ,g n be any sequence in G . Since G = HA we have ... that g( = at hj where a, € A, ht € H. Hence [ a, g 1 ...... ,9 n ] = [ a , a ^ ...... anhn ] Now [ a, a^h^ ] = [ *, h^ ] [ a, a^ ] 1 by simple commutator calculus - [ a , h1 ] since A g ©Jj Now A < G and so [a , h ]gA. Hence by a simple induction we have [ a , , . , . . ,h^ ] = [ a, ]• But by hypothesis [ a, h1 ...... h^ ] £ H and since A £ a , , ...... h^ ] € A . So £ a, h^ , .... ,h^ ] € a n h = 1 . So £ a, g1, .. . . ,gn ] = 1 for all sequences g 1 * .....g r). Hence a g r ( G ) and since a was arbitrary A s £ ^ (G ). n N o w since • H C t ^ « H C l+i H < H C, (G ) < • • . <3 H C n ( G ) = G . . n l»c* H ^ G • We can now easily prove theorem G,1 with the aid of the le m m a . Proof (of theorem G.1) We use induction on d. If d = 1 then G is abelian and H is obviously a normal subgroup. Hence assume d > 1 and that the theorem is true for soluble groups of derived length less than d. Let A be the last but one term of the derived series of G . Then A < G and A ( ffl . Since the hypotheses of the theorem remain true on taking homomorphic images of G , we have by induction on d, since G / A has derived length d-1, that H A / A <3nCd" ’ ) G /A « u a n (d - ) _ i . e . H A But the group HA satisfies the hypotheses of lemma 4.1 and so by the lemma H n H A nd Hence H We prove theorem H . 1 in a fairly similar way using the following lemma in place of lemma 4.1. Lemma 4,2 Let G = HA where A H ^ ,....,H ^ of generating sets of H such that [ g , H 1 ...... ] s H, then H Proof As in lemma 4.1 we have H n A o G and since our hypotheses remain true under homomorphic images of G we may factor by H p A and assume H p A = 1. Let a g A. Then by hypothesis, for a sequence ,.... ,H ^ of generating sets of H we have X [ H 1# ...... H n ^ S ince A < G we have [ a , H 1 ( ....,H n ] s A and so S H n A = 1 . [ a» * • • •' ,Hn 3 Hence [ a, H H ^ ] centralizes all elements in a generating set of H and so is contained in the centralizer of H. Now suppose that [ a, ,.. •. *Hn_j ] s C anc* ll+1 € f a ' H l , " , , , H n- As before if g = h a' some h € H. a' € A [ il+1. a ] - C It+1 * h 3 Since we supposed [ a, ,.... ] s wc have [ 1|+1, g ] € Ct Hence l J+1 € C1+1 By induction on i we obtain A s Cn(G ). As before, since H Ct(G ) H C i t , ( G J . Ct(G ) ^ C , C O ) with r „(G)- 1 , wo have * o »> .SlunUMIWUMSKE H Hence H «3 G . Proof of Theorem H . 1 To show sufficiency of the condition we use induction on d, the derived length of G . If d = 1 then G is abelian and H n(d- 1 ) a h / a V a Hence H A <3 ^ G . Since the group H A satisfies the conditions of lemma 4.2 we obtain H To show the necessity of the condition let H sn G and have normal closure series H = Ln < Ln_i o • • • o L 0 = G Th e n if H ...... H are generating sets of H and 1 m g g G we have [ 9 , H 1 ...... Hm ] s L m 0 s m s n. Sec. 2 Generalized Soluble Groups For SI* - groups we obtain the following two theorems analogous to theorems G.1 and H.1 in section 1 of this chapter. >1 Theorem G.2 Let G be an SI* - group, and let H be a subgroup of G Suppose that for each g g G and for all sequences h^ ,., of H that there exists an integer n = n (g ) such that [ g , , ,h n ] € H . T h e n H is ascendant in G . Corollary G.2 Let G be an SI* - group and let H be a subgroup of G Then H asc < H, g > yggG im p lie s that H asc G. Theorem H .2 Let G be an SI* - group and let H be a subgroup of G Then if for each g g G there exists an integer n = n(g) and a sequence of generating sets of H such that [ g H ,H T s H then 1 ’ n J H asc G . We mention that as shown in [i©] the condition stated in theorem H .2 is not necessary for ascendancy, for let G be the wreath product C\B of a group C of type by an elementary abelian group of order 4 generated by i and J. Then G Is locally nilpotent and satisfies the minimal condition, and hence is hypercentral. Let H = < C , l> , then H asc G since G ts hypercentral Now for any sequence ,., .. , of generating sets of H we have that ,H I where [ [ J• c ]n_ 1 l ] € [ J* n J c Is any generator of C. By a simple induction we can show that n-2 _ ^ - 2 ^ - 2 ¿ 1-B _ ^ - 2 (- 1 ) ( c c J c c J ) c r J’ C )n - 1 1 3 ^ - 2 But if this element lies In H we must have c However for eaqlp n we can always find a generator c 1 such that c 1 / 1. Hence we can not find a sequence of generating sets of H such that [ j, H , ] s i- We note, however, that if H is finitely generated then the condition is necessary [ See [ 10] lemma 2.4 ]. We begin the proof of theorem G.2 with the following lemma. * Yc^C^-W * Yr/H>*