d-Wise generation of prosolvable groups

Mathematische Nachrichten

Moscatiello

2018

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Denote by = ( ) and = ( ), respectively, the smallest and the largest cardinality of a minimal generating set of a finite group . The Tarski irredundant basis theorem implies that for every with ( ) ≤ < ( ) there exist a minimalagain a minimal generating set of . In this case we say that̃is an immediate descendant of . There are several examples of minimal generating sets of cardinality smaller than ( ) which have no immediate descendant and so it appears an interesting problem to investigate under which conditions an immediate descendant exists. In this paper we discuss this problem in the case of finite soluble groups. K E Y W O R D S generating sets, soluble groups, Tarski theorem M S C ( 2 0 1 0 ) 20F05 1022

“…Put

F = {End}_{H} false(V false)

. Proposition [, Proposition 2.1] Let

H = ⟨ h_{1}, \dots, h_{t} ⟩

and

w_{i} = false(v_{1, i}, \dots, v_{δ, i} false) \in V^{δ}

with

1 \leq i \leq t

. The following are equivalent.…”

Section: Preliminariesmentioning

confidence: 99%

The Tarski irredundant basis theorem and the finite soluble groups

Mathematische Nachrichten

Moscatiello

2018

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“…, a 5 ) generates V 5 as an H-module and {z, (1,2,3,4,5), (1, 2, 3)(4, 5, 6, 7)} is a CIG-set of G. Thus G is a (non-soluble) CIG group. Note that in this case the trivial element e is contained in the CIG-set {e, (1,2,3,4,5), (1, 2, 3)(4, 5, 6, 7)} of H, and dim F C V (e) = 5, so that the condition of Theorem 5 is satisfied. 2 Theorem 5 gives a necessary condition on u in order to ensure that the semidirect product G = V u H is a CIG group given that the irreducible linear group H is CIG.…”

Section: Cig and Pcig Groupsmentioning

confidence: 99%

Invariable generation with elements of coprime prime-power orders

Detomi

2015

Journal of Algebra

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A finite group G is coprimely-invariably generated if there exists a set of generators {g1,…,gd} of G with the property that the orders |g1|,…,|gd| are pairwise coprime and that for all elements x1,…,xd in G the set {g1^x1,…,gd^xd} generates G. In the particular case when |g1|,…,|gd| can be chosen to be prime-powers we say that G is prime-power coprimely-invariably generated. We will discuss these properties, proving also that the second one is stronger than the first, but that in the particular case of finite soluble groups they are equivalent

“…The generating graph

Γ (G)

of a finite group G is the graph defined on the elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G . It was defined by Liebeck and Shalev in [22], and has been further investigated by many authors: see for example [4–8, 10, 20, 25, 27, 28, 31] for some of the range of questions that have been considered. Many deep structural results about finite groups can be expressed in terms of the generating graph, but of course

Γ (G)

encodes significant information only when G is a 2‐generator group.…”

Section: Introductionmentioning

confidence: 99%

Graphs encoding the generating properties of a finite group

Acciarri

Mathematische Nachrichten

2020

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Assume that G is a finite group. For every a,b∈N, we define a graph normalΓa,bfalse(Gfalse) whose vertices correspond to the elements of Ga∪Gb and in which two tuples (x1,⋯,xa) and (y1,⋯,yb) are adjacent if and only if ⟨x1,⋯,xa,y1,⋯,yb⟩=G. We study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and we investigate the relations between their properties and the group structure, with the aim of understanding which information about G is encoded by these graphs.