1,721,186 research outputs found
The virtually generating graph of a profinite group
Full text linkWe consider the graph whose vertices are the elements of a finitely generated profinite group G and where two vertices x and y are adjacent if and only if they topologically generate an open subgroup of G. We investigate the connectivity of the graph obtained from by removing its isolated vertices. In particular, we prove that for every positive integer t, there exists a finitely generated prosoluble group G with the property that has precisely t connected components. Moreover, we study the graph, whose vertices are again the elements of G and where two vertices are adjacent if and only if there exists a minimal generating set of G containing them. In this case, we prove that the subgraph obtained removing the isolated vertices is connected and has diameter at most 3
Applying extremal graph theory to a question on finite groups
No full textA word w∈ Fr is said to satisfy a probability gap if there exists a constant δw< 1 such that, for any finite group G, if the probability that w(g1, g2, ... , gr) = 1 in G is at least δw, then w is an identity in G. Moreover we saythat a group G has the wt-property if however r subsets X1,.. , Xr of G are chosen with | X1| = ⋯ = | Xr| = t, there exists (g1, ... , gr) ∈ X1× ⋯ × Xr such that w(g1, ... , gr) = 1. We prove that if w satisfies a probability gap, then for every positive integer t there exists a constant ct such thatif a finite group G satisfies the wt-property, then | G| ≤ ct or w is an identity in G
ANSWERING A QUESTION ON EQUALLY COVERED GROUPS
No full textFoguel, Moghaddamfar, Schmidt, and Velasquez-Berroteran asked in [Int. Electron. J. Algebra, 37(2025), 352-365] whether there exists a positive integer n with the property that, for every finite group G, the Cartesian power Gn can be expressed as the union of a family of proper subgroups of the same order. We prove that the answer is negative
Recognizing a finite group from the generating properties of its subsets
No full textWe assume to have information about the generating properties of the subsets of a finite group G. In particular, we consider the two following situations. We know, for every subset X of G, whether X is a generating set of G. We know the graph whose vertices are the subsets of G and in which there is an edge connecting X and Y if and only if X∪ Y is a generating set of G. We discuss how this kind of information can be used to discover properties of the group G
Solubilizers in profinite groups
Full text linkThe 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 x 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
The independence graph of a finite group
Full text linkGiven a finite group G, we denote by Δ (G) the graph whose vertices are the elements G and where two vertices x and y are adjacent if there exists a minimal generating set of G containing x and y. We prove that Δ (G) is connected and classify the groups G for which Δ (G) is a planar graph
Finite groups with planar generating graph
Full text linkGiven a finite group G, the generating graph Γ(G) of G has as vertices the non-identity elements of G, and two vertices are adjacent if and only if they are distinct and generate G as group elements. Let G be a 2-generated finite group. We prove that Γ(G) is planar if and only if G is isomorphic to one of the following groups: C2, C3, C4, C5, C6, C2 × C2, D3, D4, Q8, C4 × C2, D6
Minimal generating sequences of F-subgroups
Full text linkThe behaviour of generating sets of finite groups has been widely studied, from several points of view. The purpose of this note is to investigate what happens when, instead of sets of elements generating a group, sets of subgroups belonging to a prescribed family are considered. Some known results on generating set can be extended and generalized, using similar arguments and techniques, but interesting open questions also arise
Maximal Intersections in Finite Groups
Full text linkFor a finite group G, we investigate the behavior of four invariants, MaxDim (G) , MinDim (G) , MaxInt (G) and MinInt (G) , measuring in some way the width and the height of the lattice M(G) consisting of the intersections of the maximal subgroups of G
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