Finite groups that need more generators than any proper quotient

Volta, Francesca Dalla; Lucchini, Andrea

doi:10.1017/s1446788700001312

Cited by 47 publications

(39 citation statements)

References 14 publications

(16 reference statements)

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“…The structure of a ®nite group with this property is described in [1], Theorem 1.4: there exist a ®nite group L On the minimal number of generators of free pro®nite products of pro®nite groups 55…”

Section: The Main Resultsmentioning

confidence: 99%

On the minimal number of generators of free profinite products of profinite groups

Lucchini¹

2001

Journal of Group Theory

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The solubilizer of an element x of a profinite group G is the set of the elements y of G such that the subgroup of G generated by x and y is prosoluble. We propose the following conjecture: the solubilizer of x in G has positive Haar measure if and only if g centralizes 'almost all' the non-abelian chief factors of G. We reduce the proof of this conjecture to another conjecture concerning finite almost simple groups: there exists a positive c such that, for every finite simple group S and every (a, b) ∈ (Aut(S) \ {1}) × Aut(S), the number of s is S such that a, bs is insoluble is at least c|S|. Work in progress by Fulman, Garzoni and Guralnick is leading to prove the conjecture when S is a simple group of Lie type. In this paper we prove the conjecture for alternating groups.

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Section: The Main Resultsmentioning

confidence: 99%

On the minimal number of generators of free profinite products of profinite groups

Lucchini¹

2001

Journal of Group Theory

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“…Suppose that some Sylow subgroup P of G is non-cyclic: then P ≤ N . We then have P H is a quotient of G and so is 2-generator and by [10] requires more than 2 generators, a contradiction. It follows that all Sylow subgroups of G are cyclic and G ∈ D.…”

Section: Corollary 3 Let T 0 Be the Class Of T-groups G With G/g Cyclmentioning

confidence: 92%

Graphs, partitions and classes of groups

Ballester‐Bolinches

Cossey

2010

Monatsh Math

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If X is a class of groups, Delizia et al. (Bull Austral Math Soc 75:313-320, 2007) call a group G X-transitive (or an XT -group) if whenever a, b and b, c are in X a, c is also in X (a, b, c ∈ G). The structure of XT -groups has been investigated for a number of classes of groups, by Delizia, Moravec and Nicotera and others. A graph can be associated with a group in many ways. Delizia, Moravec and Nicotera introduce a graph which is a generalisation of the commuting graph of a group, but do not make use of the graph. We will use the properties of the graph to investigate further classes of groups and to obtain more detailed structural information.

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“…The following lemma is a sharpening of Lemma 1 of [3]. The proof is very technical and follows the same lines as the proof of Theorem 1.1 in [7] but at each step we take care of lifting generators in as many ways as we can.…”

Section: Background Materialsmentioning

confidence: 99%