Bases of certain finite groups

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Introduction
Recently, considerable attention has been paid to problems on generators of groups.This concerns e.g.small generating sets of finite simple groups (see the survey [2]).Generating sets of groups are also relevant from the graph- theoretical point of view.because they give rise to connected Cayley digraphs (see e.g.[3] ). Furthermore.Beneteau [1] revisited Burnside basis theorem, by restating it in terms of matroids.Properties of this kind have also been studied by Jones [5] for semigroups, often reducing questions from semigroups to groups.Therefore, it seems to be interesting to investigate groups whose behaviour with respect to generators is fairly similar to that of p-groups.This study can be performed with the help of an 'abstract* notion of independence.In order to make this idea more precise, we recall the definition of a matroid.Let S be a set, and let c : P(S) --~ P(S) be a mapping satisfying, for all X C S and all The pair (S. c) is then called a matroid.This is one of the many.equivalent definitions of a matroid (see e.g.[6]), and it seems to be the more appropriate for our purpose...B subset .~' of S is said to be c-independent if c(~' j ~ c(.x ) for all Y C X. We say that X is a generating set if c(X) =6*.Also, a basis is an independent generating set.We will make use of the above definitions even for the more general case of a mapping c satisfying only (1)-(3).For instance, let G be a finite group, and let be its Frattini subgroup (i.e. the intersection of its maximal subgroups).Setting c(X) = X, 03A6(G) > we obtain such a mapping.Following ~7~, [8], a subset .~' of a group is said to be independent if it is c-independent with respect to this c.Accordingly with our previous paper [7], a finite group G, is said to be a matroid group if it satisfies the following two axioms: (Ml) the minimal generating sets of G are exactly the bases of a matroid ~~~ (G,): , (~I2) each independent subset ~' of G is contained in a minimal generating :setofG:' :- .. ' " z.: . Note that in group theory the terms 'basis' and 'minimal generating set' are synonymous.In this paper we will use the latter, in order to avoid confusion with the 'basis' of a matroid.
As a consequence of Burnside basis theorem, all finite p-groups are matroid groups.We studied these groups in ( ~j , [8], obtaining a characterization for them ((7~.Lemma 1.1 and Theorem 2.5), that we summarize below in the following ~ way.(Here Op(G) is the maximum normal p-subgroup 'of G).
Theorem 1.1 Let G be a finite group, and let H = G/03A6(G) .Then G is a matroid group if and only if one of the following holds: (1) G is a p-g~roup: . ' ° there are two pri-mes p. q with p ~ l. (mod q) such that |H : Op(H)| = q.Op(H) is elementary abelian and every subgroup of Op(H) is normal (but not central) in H .By the above, all matroid groups are solvable.But a finite simple group cannot even satisfy (M1), because its minimal generating sets have different numbers of elements.The situation is less obvious for (VI2).We are aware of no non-solvable finite group satisfying this condition, and we conjecture that no such group exists.However, the solvable case is solved by Theorem 1.2 below.
(Here F(G ) is the Fitting subgroup of G, i.e. its maximum normal nilpotent subgroup).
Theorem 1.2 Let G be a finite solvable group, and let H = G/03A6(G).Then G satisfies (1~~2) if and only if one of the following holds: (1) H is abelian ; (2) F(H) ) has prime index in H and all its subgroups are nor:mal in H.
Another characterization of finite matroid groups will be given by using the following concept.Let (S. c) be a matroid and let ~ C S. Then...)[ is said to be c-hierarchical if its elements can be ordered in a sequence x~, x~~, ..., xm such that c(x1, for all k = 2, ..., m, and c(~) ~ For the case of groups (with c(.~' ) _ .~,'> ), this concept was introduced by Hamidoune, Llado and Serra, who found in [4] the values of connectivity of Cayley digraphs associated with hierarchical generating sets.
At a first glance, most finite groups have hierarchical generating sets that are not minimal.On the other hand, we are able to describe the groups all whose hierarchical generating sets are minimal.This is the content of the following theorem, which will be proved as a corollary of a slighty more general result (Theorem 3.2 ).
Theorem 1.3 Let G be a finite group.Then all hierarchical generating sets of G are minimal generating sets if and only ifC is a 'matroid group with = I.
The rest of the paper is mostly devoted to the proof of Theorems 1.2 and 1.3.Terminology and notation are standard, and can be found in [9] (but we write 03A6(G) and F(G) for the Frattini and Fitting subgroups).we will also use some obvious shortenings, such as .~'B x for ~' B ~x }. 2 Groups satisfying (M2) First of all, we want to prove Theorem 1.2.Note that a group G satisfies (NI2) if and only if G/ ~ (G' ) does.Hence, in Lemmas 2.1-2.4,as w ~ell as in the proof ' of the theorem, we will assume that G' is a finite solvable group satisfying (M2) and~(G)=1.. PROOF.As 03A6(G) = 1, Op(G) is elementary abelian.Let X be a minimal generating set of Op(G'), and let :~' U V be a minimal generating set of G, Then X U y, is independent for all y E r' .~ Hence for each x E .~' ,.~'B > is a subgroup y >, maximal of index p.Thus .~'Bx > !.~'Bx: y >, .so that :'~' B x > 4G for all x E':~'.Since is the direct product of the Op(C).we are done./ Lemma 2.2 The group G is supersolvable and metabelian.Also, G' F(G).
PROOF.We know that F(C) .isabelian and all its subgroups are normal in G.
Therefore, by conjugation, G induces on F(G) universal power-automorphisms, that of course commute each other.Then this action corresponds to a homomor- phism whose kernel is CG (F (G ) ) and whose image is abelian, so G/C'c(F(G)) is abelian.By [9], 7.4.7,p.187, G/CG(F(G)) = F(G), then G/F(G) is abelian.Hence G' ~ F(C).and G is metabelian.Moreover it is obviously supersolvable because every subgroup of G' is normal in C, so that G has a chief series with factors of prime order.[] Lemma 2.3 Every element of G B F(C) belongs to a complement of F(G).
. By contradiction, suppose that can be embedded in a minimal genera- .ting set.Then, in particular, there is z % jFf such that X'~{y, z} is independent.
In view of Lemma 2.3.there is a complement M of F (G) containing :. Suppose that Cc(~).From Lemma 2.2 it follows that M n G' = 1, so M is abelian.
Furthermore H M = C, and so = G.Therefore z = Z(C).But by [9], 7.4.7,p. 167 H, a contradiction.Then there is w 6 X' ~ y such that [w, z] ~ ' By Lemma 2.2 we have G' ~ F(G), so we can choose w = y and then there is xk ~ X that may be replaced by a power of [y, z] having prime order.This gives rise to another minimal generating set of F(G).
Assume now that (2) holds.Let X be independent and such that H = .k'> is different from G, and prove that there is y ~ X such that X ~ y is independent, too.
If F(C) is not contained in H, choose any y E F(G) B H. Now A U y is .independent.In fact, for all x E .'~'w'e have .~'~ x, y >== .~C~ x > y >~ ~ >. as AB.r >~ ~.
, If F ( G ) is contained in H then H = F (G ). because F (G is maximal in G. Let y E G B F(G).Clearly, the set .~' U Y generates G.Moreover, for all j? E .~',K = .~'B :~ > F (G' ), and so K j G'.It follows that ~' ~ x, ;y ~= K y >, which is a proper subgroup of G.This proves that .xU Y is independent, and so G satisfies (1~~I2). 1   .By Theorem 1.2, an elementary example of a nonabelian group satisfying ('~i2) is given by any dihedral group.
It is w orth to mention some consequences of Theorem 1.2.Corollary 2.5 Let G be a solvable group satisfying (M2), and let .4be an abelian> group with (|A|.|F(G)| ) = I .Then G x .4satisfies (M2).Corollary 2.6 Let G be a solvable group satisfying (M2), let p be the index of F(G ) in G. Then the Sylow p-subgroup of F(G') ) is a direct factor of G. Also, if G has no abelian direct factors.then F(G) is a Hall subgroup of G.
. PROOF.Let P be a p-Sylow subgroup of G. Since Q = and P is abelian, we have [Q Yet another easy consequence of Theorem 1.2 is the following.Corollary 2.7 .4 finite solvable group G satisfying (M2) is a matroid group if and only if F{G)/~{G') is a p-group.

'
The following proposition provides a criterion to prove that a group does not satisfy (M2).Note that a group does not satisfy (M2) if and only if it contains a maximal independent set that is not a minimal generating set.Proposition 3.4 If G is an F-matroid group and F H G, then G is also an H-matroid group.
Suppose, by contradiction, that {y1, ..., yn} is not H-independent.The converse of Proposition 3.4 is not true.Obvious counter-examples arise from the fact that if H is a maximal subgroup of a group G, then G is always an H-matroid group.