Adolfo Ballester-Bolinches and Tatiana Pedraza
This paper is devoted to the study of groups G in the universe c¯
of all radical locally finite groups with min-p for all primes p suchthat every δ-chief factor of G is either a cyclic group of prime orderor a quasicyclic group. We show that within the universe c¯
class of groups behaves very much as the class of finite supersolublegroups.
A group G is said to be supersoluble if it has a finite normal series of
cyclic factors. A chief factor of a supersoluble group is cyclic of primeorder and a maximal subgroup has prime index. In fact these two prop-erties characterize finite supersoluble groups. It is well known that inthe universe of all finite groups the class of supersoluble groups forms asubgroup-closed saturated formation which is intermediate between theclasses of nilpotent groups and soluble groups. Several authors have in-vestigated supersoluble groups, not necessarily finite, and generalizations(hypercyclic groups and locally supersoluble groups) extending some re-sults of finite supersoluble groups and establishing connections betweenthese classes of generalized supersoluble groups.
In this paper, we introduce a class of generalized supersoluble groups
L of all radical locally finite groups with min-p for all
primes p: the class U ∗. It is intermediate between the classes of super-soluble c¯
L as supersoluble groups do in the class of all finite groups.
2000 Mathematics Subject Classification. 20F16, 20F19, 20F50. Key words. Locally finite groups, generalized supersoluble groups, major subgroups. This research is supported by Grant BFM-2001-1667-C03-03, MCyT (Spain) andFEDER (European Union).
The main purpose of this section is to establish the notation, ter-
minology and some results which will be used throughout this paper. Notation that is not specifically cited here is consistent with that usedin [7], [8] and [10].
An infinite group can have insufficient maximal subgroups or even
none at all. In order to avoid this situation, Tomkinson [11] introducesthe notion of major subgroup. We recall this definition. Let U be asubgroup of a group G and consider the properly ascending chains
U = U0 < U1 < · · · < Uα = G
from U to G, then m(U ) is the least upper bound of the types α of allsuch chains. Clearly m(U ) = 1 if and only if U is a maximal subgroupof G.
A proper subgroup M of G is said to be a major subgroup of G
if m(U ) = m(M ) whenever M ≤ U < G. This is a nice extension of theconcept of maximal subgroup in the sense that every proper subgroup ofa group G is always contained in a major subgroup of G [11, (2.3)]. Inparticular, the intersection of all major subgroups of a group G, denotedby µ(G), is a proper subgroup of G with properties similar to those ofthe Frattini subgroup of a finite group.
In the sequel, we tacitly assume that all groups belong to the class c ¯
of all radical locally finite groups with min-p for all primes p.
Let G be a group and let M be a major subgroup of G. Denote
MG = CoreG(M ). Then G/MG is either a finite soluble primitive group,if M is a maximal subgroup of G, or a semiprimitive group, if M is notmaximal in G (here, a group G is said to be semiprimitive if it is the splitextension, G = [D]M , of a faithful divisibly irreducible ZM -module Dby a finite soluble group M ). This result, proved in [2], confirms theimportance of major subgroups in the study of the structure of groupsand motivates some definitions which are in some sense extensions ofwell known ones in the finite universe.
Definition 1. Suppose that G is a group and let M be a major subgroupof G. We define
Soc(G/MG), if M is a maximal subgroup of G
if M is not a maximal subgroup of G.
A Class of Generalized Supersoluble Groups
In both cases, DM /MG = F (G/MG), (DM /MG) ∩ (M/MG) = 1 and
M /MG) = DM /MG for every major subgroup M of G.
Let G be a group and consider two normal subgroups H and K of G
such that K is contained in H. Then H/K is called a δ-chief factor of Gif H/K is either a minimal normal subgroup of G/K or a divisibly irre-ducible ZG-module, that is, H/K has no proper infinite G-invariant sub-groups. Every δ-chief factor is either an elementary abelian finite p-groupfor some prime p or a direct product of finitely many quasicyclic p-groupsfor some prime p (see [7, (1.2.4)]).
Let G be a group. We say that G is a U ∗-group if every δ-chief
factor of G is either a cyclic group of prime order or a quasicyclic group. Obviously the class U ∗ is a class of generalized supersoluble groups inthe universe c¯
L because every finite U ∗-group is supersoluble and every
L-group is finite and so it is a U ∗-group.
On the other hand, since every group contains minimal normal sub-
groups it follows that every U ∗-group is hypercyclic (and hence locallysupersoluble [1]). Moreover the class U ∗ is intermediate between theclasses of supersoluble groups and hypercyclic groups. On one hand,every quasicyclic p-group, p a prime, is a non-supersoluble U ∗-group. On the other hand, let G be a cyclic group of order 22. Consider adivisibly irreducible ZG-module A, faithful for G, such that A is a peri-odic 2-group (A always exists and it is unique up to isomorphism by [9,(3.5)]). Then, applying [9, (5.9)], A has rank 2. In particular, theChernikov group X = AG is not in the class U ∗. Moreover, if K is anormal subgroup of X such that X/K = 1 then X/K contains a min-imal normal subgroup N/K. Since X/K is a 2-group then it is locallynilpotent and then, by [7, (1.2.6)], N/K is a cyclic group of order 2. Weconclude that X is a hypercyclic group.
Let B be the class of all groups in which every proper subgroup has
a proper normal closure. The class B has been introduced and studiedin [3], [4]. The results of these papers show that this class is intermediatebetween nilpotent and locally nilpotent groups, and that it is the naturalgeneralization of the class of finite nilpotent groups from the finite uni-verse to the universe c¯
L. We show that the largest normal B-subgroup
of a group is the Fitting subgroup and every δ-chief factor of a group Gin B is central in G. Consequently, every δ-chief factor of G has rankone and G is a U ∗-group. Therefore, we obtain in the universe c ¯
inclusions to the finite universe for these classes of generalized nilpotent
groups and generalized supersoluble groups:
{Abelian groups} ⊂ {Nilpotent groups} ⊂ B ⊂ U ∗
and all these inclusions are proper.
Let p be a prime. We say that a group G is a Bp-group if G is
p-nilpotent and the Sylow p-subgroups of G are nilpotent. The class Bpis a local version of the class B. The results of the paper [5] showthat Bp is a subgroup-closed formation which plays the same role in theuniverse c¯
L as finite p-nilpotent groups do in the finite one. In partic-
ular, every group G has a unique largest normal Bp-subgroup denotedby δp p(G), for every prime p, which is the intersection of the centralizersof all δ-chief factors of G which are p-groups and F (G) =
Let G be a U ∗-group and let p be a prime. If H/K is a δ-chief factor
of G which is a p-group then G/ CG(H/K) is a cyclic group of orderdividing p − 1 if p = 2 or a cyclic group of order dividing 2 if p = 2 by [7,(1.5.18), (1.5.19)]. Therefore G/δp p(G) is in the class A(p − 1) if p = 2or in the class A(2) if p = 2, where A(n) denotes the class of all abeliangroups with exponent dividing n. Moreover G/Op p(G) ∈ A(p − 1) forevery prime p because Op p(G) is the intersection of the centralizers ofall p-chief factors of G by [7, (6.2.4)].
A finite group G is supersoluble if and only if for every prime p,
G/Op p(G) is abelian of exponent dividing p − 1. If G is a c¯
that condition does not imply that G ∈ U ∗ (consider for instance theexample of a 2-group in Section 2 which is not a U ∗-group). Our firstresult provides a necessary and sufficient condition for a group G to bea U∗-group.
Theorem 1. A group G is a U ∗-group if and only if for every prime p,G/δp p(G) is in the class A(p − 1) if p = 2 or in the class A(2) if p = 2,where A(n) denotes the class of all abelian groups with exponent divid-ing n.
Proof: Let G be a group such that for every prime p, G/δp p(G) is inthe class A(p − 1) if p = 2 or in the class A(2) if p = 2. Consider H/Ka chief factor of G. In particular H/K is a p-group for some prime p. Since G/ CG(H/K) is in the class A(p − 1) if p = 2 or in the class A(2)if p = 2 it follows from [8, B, (9.8)] that H/K is a cyclic group oforder p. Suppose now that H/K is a divisibly irreducible ZG-module. In particular H/K is a p-group for some prime p. Suppose that p = 2.
A Class of Generalized Supersoluble Groups
Then L = G/ CG(H/K) is in the class A(p−1). Since H/K is a divisiblyirreducible ZL-module which is faithful for L it follows from [9, (3.1)]that L is a cyclic group. Since |L| divides p − 1, it follows from [9, (3.4)]that H/K has rank 1, that is, H/K is a quasicyclic p-group. Assume nowthat p = 2. Then L = G/ CG(H/K) is in the class A(2). Applying [9,(3.1)] we have that L is trivial or a cyclic group of order 2. ConsequentlyH/K has rank 1 by [9, (3.4)]. We conclude that G is a U ∗-group.
Our next results analyze the behaviour of U ∗ as a class of groups
and they are motivated by the well known fact that, in the finite uni-verse, the class of all supersoluble groups is a subgroup-closed saturatedformation [8, VII, (2.19)].
Recall that a class F of groups is said to be a formation if it satisfies
1. If G ∈ F and N is a normal subgroup of G, then G/N ∈ F. 2. If {Ni}i∈I is a collection of normal subgroups of G such that
Theorem 2. The class U ∗ is a subgroup-closed formation.
Proof: It is clear that U ∗ is closed under taking epimorphic images, thatis, U∗ is Q-closed.
Let {Ni}i∈I be a collection of normal subgroups of a group G such
we have that G/Ni/(δp p(G/Ni)) is in the class A(p − 1) if p = 2 orin the class A(2) if p = 2, for all i ∈ I. Let us denote by GA(n)the A(n)-residual of a group G.
GA(p−1)Ni/Ni = (G/Ni)A(p−1) ≤ δp p(G/Ni) for all i ∈ I. In par-ticular, GA(p−1)Ni/Ni is in the class Bp for all i ∈ I. Since Bp isa formation by [5, Theorem 1] we obtain that GA(p−1) ∈ Bp. Con-sequently G/δp p(G) ∈ A(p − 1). Suppose now that p = 2. ThenGA(2)Ni/Ni = (G/Ni)A(2) ≤ δ2 2(G/Ni) for all i ∈ I. In particular,GA(2)Ni/Ni is in the class B2 for all i ∈ I. Therefore GA(2) ∈ B2and hence G/δ2 2(G) ∈ A(2). It follows from Theorem 1 that G ∈ U ∗. Consequently U ∗ is a formation.
We prove now that U ∗ is subgroup-closed. Let G be a U ∗-group and
let H be a subgroup of G. We have to prove that H is also a U ∗-group. We split the proof into three cases:
Case 1: G is a Chernikov group such that G0 (the radicable part of G) isa quasicyclic p-group for some prime p. Since H 0 is a divisible subgroupof G0, then either H0 = 1 or H0 = G0. First we assume that H0 = G0.
Let A/B be a δ-chief factor of H. Since the class U ∗ is Q-closed, we mayassume that B = 1. If A is a divisibly irreducible ZH-module, then A is aquasicyclic p-group. Suppose now that A is a minimal normal subgroupof H. Then either A ≤ H0 or A ∩ H0 = 1. If A is contained in H0,then A is a cyclic group of prime order p because it is an elementaryabelian. Assume that A ∩ H0 = 1. Since AH0/H0 is a minimal normalsubgroup of H/H0 and H/H0 is a finite supersoluble group, it followsthat AH0/H0 is a cyclic group of prime order and so is A. Suppose nowthat H0 = 1, that is, H is finite. If A/B is a chief factor of G, then A/Bis a cyclic group of prime order. In particular (H ∩ A)/(H ∩ B) iseither trivial or a cyclic group of prime order. Hence a chief series of Gintersected with H will yield, after deleting redundant terms, a chiefseries of H (of finite length) with cyclic factors. Therefore H ∈ U ∗.
Case 2: G is a Chernikov group with Op (G) = 1 for some prime p. ThenG = G0M , where M is a finite subgroup of G. We can certainly assumethat G0 = 1, since otherwise G is a finite supersoluble group and so theresult follows. Therefore G0 is a divisible abelian p-group of finite rank. By [12, (1.3)] there is a finite normal subgroup C of G contained in G0such that G0/C is a direct product of divisibly irreducible ZG-modules,say
G0/C = (G1/C) × (G2/C) × · · · × (Gn/C).
Since Gi/C is a δ-chief factor of G and G ∈ U ∗, we have that Gi/C is aquasicyclic p-group for all i ∈ {1, . . . , n}. Let us first assume that C = 1and denote X1 = G2 × G3 × · · · × Gn and Xi = G1 × · · · × Gi−1 × Gi+1 ×· · · × Gn for all i > 1. It is clear that Xi is normal in G for all i and
p-group. Applying Case 1, we have that H/(H ∩ Xi) is a U∗-group forevery i. Since U ∗ is a formation, we conclude that H ∈ U ∗. Assume nowthat C = 1. By the above argument, HC/C is a U ∗-group. We provethat HC ∈ U∗. Let A/B be a δ-chief factor of HC. There is no loss ofgenerality in assuming that B = 1. Suppose that A is a minimal normalsubgroup of HC. Then either A ∩ C = 1 or A ∩ C = A. If A ∩ C = 1,it follows that AC/C is a minimal normal subgroup of HC/C ∈ U ∗ andso A is a cyclic group of prime order. Assume that A ≤ C. Let
be part of a chief series of G passing through C, that is, Ci/Ci−1 is aminimal normal subgroup of G/Ci−1 for all i ∈ {1, . . . , s}. Since G ∈ U∗,we have that Ci/Ci−1 is a cyclic group of prime order for all i. On theother hand, since 1 = A ≤ C there exists j ∈ {1, . . . , s} such that
A Class of Generalized Supersoluble Groups
Cj−1. In particular, 1 = ACj−1/Cj−1 ≤ Cj/Cj−1
= ACj−1/Cj−1 is a cyclic group of prime order. Suppose
now that A is a divisibly irreducible Z(HC)-module. Then A ∩ C is aproper finite subgroup of A. This implies that A/(A ∩ C) ∼
divisibly irreducible Z(HC/C)-module and so it is a quasicyclic group. Consequently A is a quasicyclic group. We have proved that HC is aU∗-group. Since U ∗ is a formation and C is an abelian normal subgroupof HC, it follows from [2, Lemma 2] that H ∈ U ∗, which is our claim.
General case: By [7, (2.5.13)], G/Op (G) is a Chernikov U ∗-group forevery prime p. Applying Case 2, we have that H/(H ∩ Op (G)) is a U∗-group for every prime p. Since
we conclude that H belongs to U ∗, which completes the proof.
Bearing in mind that the class of supersoluble groups in the finite
universe is saturated, one can wonder if U ∗ could enjoy this property inour universe. The answer is negative as the following example shows:
Example 1. Let X be the regular wreath product of a quasicyclicp-group and a cyclic group of order p, where p is a prime number. Con-sider G = X/ Z(X). Then G = G0M is a semiprimitive group such thatG0 is isomorphic to a direct product of p − 1 quasicyclic p-groups andM is a cyclic group of order p. Moreover, the group G can be expressedas G =
i, where Gi = Ωi(G0)M , for each natural number i and
{Gi : i ≥ 1} is an ascending chain of subgroups of G. Notice that Giis a finite p-group and hence a supersoluble group for each i ≥ 1. How-ever, if p = 2, G is not a U ∗-group because G0 is a divisibly irreducibleZG-module which is not quasicyclic. Applying [2, Theorem A], we havethat U∗ is not saturated.
Nevertheless, U ∗ is closed under taking extensions by Tomkinson’s
Frattini-like subgroup as our next result shows.
Theorem 3. Let N be a normal subgroup of a group G such that N/µ(G)is a U∗-group. Then N ∈ U ∗.
Proof: Assume first that G is a Chernikov group. Then µ(G) is finite by[12, (1.2)]. On the other hand, for every prime p, (N/µ(G))/Op p(N/µ(G))is abelian of exponent dividing p − 1 because N/µ(G) ∈ U ∗. Moreover,the arguments used in [11, (5.1)] allow us to show that Op p(N/µ(G)) =Op p(N )/µ(G). Thus, for every prime p, N/Op p(N ) is abelian of expo-nent dividing p − 1.
Let H/K be a δ-chief factor of N . Then either Hµ(G)/Kµ(G) is
N -isomorphic to H/K or Hµ(G) = Kµ(G) and H ∩ µ(G)/K ∩ µ(G) is
N -isomorphic to H/K. Consequently, there is no loss of generality inassuming that either K ≤ H ≤ µ(G) or µ(G) ≤ K ≤ H. In the first case,we have that H/K is finite and therefore H/K is a p-chief factor of Nfor some prime p. In particular, AutN (H/K) ∼
and abelian of exponent dividing p − 1. Applying [8, B, (9.8)], H/K isa cyclic group of order p. Assume now that µ(G) ≤ K ≤ H. Then H/Kis isomorphic to a δ-chief factor of N/µ(G) ∈ U ∗ and consequently H/Kis a cyclic group of prime order or a quasicyclic group. We concludethat N ∈ U∗.
In the general case, applying [7, (2.5.13)], G/Op (G) is a Cherni-
µ(G)Op (G)/Op (G) ≤ µ(G/Op (G)) it follows that N Op (G)/Op (G)is a normal subgroup of G/Op (G) satisfying the hypothesis of the the-orem. By the above argument, we obtain that N/(N ∩ Op (G)) is aU∗-group, for every prime p. Since U ∗ is a formation, we conclude thatN belongs to U ∗, as required.
Corollary 1. Let G be a group. Then G/µ(G) ∈ U ∗ if and only if G ∈ U ∗.
If M is a major subgroup of a group G such that DM /MG is a cyclic
group of prime order or a quasicyclic group then G/MG is a U∗-group. This fact motivates the following:
Definition 2. Let G be a group and let M be a major subgroup of G. We define the extended index of M in G, denoted by qG(M ), as the rankof DM /MG.
Note that M is a major subgroup of G, then qG(M ) = 1 if and only
if G/MG is not in U∗. Consequently the following result holds. Corollary 2. Let G be a group. Then G ∈ U ∗ if and only if the extendedindex of M is 1 for every major subgroup M of G.
Note that the above result extends a well known result of Huppert [8,
VII, (2.2)]. Bathia [6] proved that the intersection of all maximal sub-groups of a finite group of composite index is a supersoluble characteristicsubgroup of the group. Our aim now is to obtain a similar result in ouruniverse.
Definition 3. Let G be a group. We define
{M : M is a major subgroup of G such that qG(M ) = 1}
{M : M is a major subgroup of G such that G/MG /
We stipulate that L(G) = G if the above set of major subgroups is
A Class of Generalized Supersoluble Groups
Theorem 4. Let G be a group. Then L(G) is a characteristic U ∗-sub-group of G.
Proof: It is clear that L(G) is a characteristic subgroup of G. We provenow that L(G) belongs to U ∗. If the above set of major subgroups isempty then L(G) = G ∈ U ∗ by Corollary 1. Then we may assumethat this set is non-empty. Assume first that µ(G) = 1. Let L(G)U∗be the U∗-residual of L(G), that is, the intersection of all normal sub-groups N of L(G) such that L(G)/N ∈ U ∗. By Theorem 1, we have thatL(G)/L(G)U∗ is a U∗-group. We will show that L(G)U∗ is a subgroupof µ(G). Obviously L(G)U∗ is contained in every major subgroup M of Gsuch that G/MG /
∈ U∗. Suppose now that M is a major subgroup of G
such that G/MG ∈ U∗. Then, the U∗-residual of G, GU∗, is containedin MG. Since U∗ is subgroup-closed, it follows that L(G)U∗ ≤ GU∗. Consequently, L(G)U∗ is a subgroup of M . This implies that L(G)U∗ iscontained in every major subgroup of G and hence L(G)U∗ ≤ µ(G) = 1. We conclude that L(G) ∈ U ∗.
Suppose now that µ(G) = 1 and consider G/µ(G). By the above
argument we have that L(G/µ(G)) ∈ U ∗. Moreover, L(G/µ(G)) =L(G)/µ(G).
Therefore it follows from Theorem 3 that L(G) is a
It is well known that if G is a finite supersoluble group, then G has a
normal Sylow p -subgroup [8, VII, (2.1)] for the smallest prime p dividingthe order of G. Moreover, the derived subgroup of G is nilpotent. Thecorresponding versions in our universe for the class U ∗ are the following:
a) G has a normal Sylow p -subgroup for the smallest prime p dividing
b) G ≤ F (G); in particular, G belongs to B.
a) Let M be a major subgroup of G and denote MG = CoreG(M ).
Suppose that the result is true for G/MG for every major sub-group M of G. Let Q be a Sylow p -subgroup of G. Since all p -sub-groups of G are conjugate and QMG/MG is a Sylow p -subgroupof G/MG by [7], it follows that G = (NG(Q))MG for every majorsubgroup M of G. Since every proper subgroup of G is containedin a major subgroup of G, it follows that G = NG(Q). That is,G contains a normal Sylow p -subgroup. Hence there is no loss ofgenerality in assuming that MG = 1 and then G is either a finite
primitive soluble group or G is a semiprimitive group by [2, Theo-rem 1]. In the first case, G is a finite supersoluble group and thenthe result is true [8, VII, (2.1)]. Consequently we may assumethat G is a semiprimitive group. Then G = [G0]M , where G0is a divisibly irreducible ZG-module, which is a q-group for someprime q, such that CG(G0) = G0 and M is a finite soluble group. Since G ∈ U∗, we have that G0 is a quasicyclic q-group. ThereforeG/ CG(G0) = G/G0 is a subgroup of a cyclic group of order q − 1if q = 2 or a cyclic group of order 2 if q = 2. Consequently pdivides |M |. If q = 2 then G is a 2-group and hence the Sylow2 -subgroup of G is trivial. Let assume that q = 2. In particularp = q. Now M is cyclic and so M has a normal Sylow p -subgroup,R say. Then Q = G0R is a normal p -subgroup of G, which is ourclaim.
b) Let H/K be a δ-chief factor of G. In particular, H/K is a p-group
for some prime p. Since G ∈ U ∗ it follows that G/ CG(H/K) is inthe class A(p − 1) if p = 2 or in the class A(2) if p = 2. There-fore G ≤ CG(H/K) for every δ-chief factor H/K of G. Since theFitting subgroup F (G) of a group G is the intersection of the cen-tralizers of all δ-chief factors of G by [3, Theorem 7], we concludethat G ≤ F (G). In particular G belongs to B.
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[6] H. C. Bhatia, A generalized Frattini subgroup of a finite group,
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A Class of Generalized Supersoluble Groups
[7] M. R. Dixon, “Sylow theory, formations and Fitting classes in lo-
cally finite groups”, Series in Algebra 2, World Scientific PublishingCo., Inc., River Edge, NJ, 1994.
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[10] D. J. S. Robinson, “Finiteness conditions and generalized soluble
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Adolfo Ballester-Bolinches:Departament d’ `
Universitat de Val`enciaDr. Moliner 5046100 Burjassot, Val`enciaSpainE-mail address: Adolfo.Ballester@uv.es
Universidad Polit´ecnica de ValenciaEscuela Polit´ecnica Superior de Alcoy03801 Alcoy, AlicanteSpainE-mail address: tapedraz@mat.upv.es

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