Connected Components in the Invariably Generating Graph of a Finite Group

Garzoni, Daniele

doi:10.1017/s0004972721000149

Cited by 3 publications

(1 citation statement)

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“…One of the first motivations for investigating finite groups having normal covering number 2 goes back to a problem in Galois theory (for more details see [8, Section 1]) and is linked to the study of intersective polynomials, that is, integer polynomials having a root modulo p, for every prime number p (see [18] and [74]). Recently, simple groups having normal covering number 2 have been used to construct sparsely connected invariably generating graphs [41]. There is also a unexpected connection between the theory of transitive permutation groups G in which every derangement is a p-element for some prime p and the normal 2-coverings of G. Indeed, as observed in [20], G has such property if and only if G admits a normal 2-covering with components given by a point stabilizer and a p-Sylow subgroup of G. As an application of our work, we give a brief proof of one of the main results in [20] in Section 11.1.…”

Section: Introductionmentioning

confidence: 99%

Normal $2$-coverings of the finite simple groups and their generalizations

Bubboloni¹,

Spiga²,

Weigel³

2022

Preprint

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Given a finite group G, we say that G has weak normal covering number γw(G) if γw(G) is the smallest integer with G admitting proper subgroups H 1 , . . . , H γw (G) such that each element of G has a conjugate in H i , for some i ∈ {1, . . . , γw(G)}, via an element in the automorphism group of G.We prove that the weak normal covering number of every non-abelian simple group is at least 2 and we classify the non-abelian simple groups attaining 2. As an application, we classify the non-abelian simple groups having normal covering number 2. We also show that the weak normal covering number of an almost simple group is at least two up to one exception.We determine the weak normal covering number and the normal covering number of the almost simple groups having socle a sporadic simple group. Using similar methods we find the clique number of the invariably generating graph of the almost simple groups having socle a sporadic simple group.

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