Generating sets of finite groups

Cameron, Peter J‎.; Lucchini, Andrea; Roney-Dougal, Colva M.

doi:10.1090/tran/7248

Cited by 11 publications

(21 citation statements)

References 25 publications

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“…The following equivalence relation ≡ m was introduced in [8, Sect. 2]: two elements are equivalent if each can be substituted for the other in any generating set for G .…”

Section: Generalizing Some Definitions and Results From [7]mentioning

confidence: 99%

“…2]: two elements are equivalent if each can be substituted for the other in any generating set for G . By [8, Prop. 2.2],

x \equiv_{normalm} y

if and only if x and y lie in exactly the same maximal subgroups of G .…”

Section: Generalizing Some Definitions and Results From [7]mentioning

confidence: 99%

“…The graphs

{normalΓ}_{1, b} false(G false)

play also a central role in the last section of the paper. In [8] an equivalence relation ≡ m has been introduced, where two elements are equivalent if each can be substituted for the other in any generating set for G . This relation can be refined to a new sequence

\equiv_{m}^{false(r false)}

of equivalence relations by saying that

x \equiv_{normalm}^{(r)} y

if each can be substituted for the other in any r ‐element generating set.…”

Section: Introductionmentioning

confidence: 99%

“…This relation can be refined to a new sequence

\equiv_{m}^{false(r false)}

of equivalence relations by saying that

x \equiv_{normalm}^{(r)} y

if each can be substituted for the other in any r ‐element generating set. The relations

\equiv_{m}^{false(r false)}

become finer as r increases, and in [8] the authors study the value

ψ (G)

of r at which they stabilise to ≡ m . Indeed results about ≡ m ,

\equiv_{m}^{false(r false)}

and

ψ (G)

can be reformulated and reinterpreted in terms of properties of the graphs

{normalΓ}_{1, b} false(G false)

.…”

Section: Introductionmentioning

confidence: 99%

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Graphs encoding the generating properties of a finite group

Acciarri

Lucchini

2020

Mathematische Nachrichten

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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.

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“…The following equivalence relation ≡ m was introduced in [8, Sect. 2]: two elements are equivalent if each can be substituted for the other in any generating set for G .…”

Section: Generalizing Some Definitions and Results From [7]mentioning

confidence: 99%

“…2]: two elements are equivalent if each can be substituted for the other in any generating set for G . By [8, Prop. 2.2],

x \equiv_{normalm} y

if and only if x and y lie in exactly the same maximal subgroups of G .…”

Section: Generalizing Some Definitions and Results From [7]mentioning

confidence: 99%

“…The graphs

{normalΓ}_{1, b} false(G false)

\equiv_{m}^{false(r false)}

of equivalence relations by saying that

x \equiv_{normalm}^{(r)} y

if each can be substituted for the other in any r ‐element generating set.…”

Section: Introductionmentioning

confidence: 99%

“…This relation can be refined to a new sequence

\equiv_{m}^{false(r false)}

of equivalence relations by saying that

x \equiv_{normalm}^{(r)} y

if each can be substituted for the other in any r ‐element generating set. The relations

\equiv_{m}^{false(r false)}

become finer as r increases, and in [8] the authors study the value

ψ (G)

of r at which they stabilise to ≡ m . Indeed results about ≡ m ,

\equiv_{m}^{false(r false)}

and

ψ (G)

can be reformulated and reinterpreted in terms of properties of the graphs

{normalΓ}_{1, b} false(G false)

.…”

Section: Introductionmentioning

confidence: 99%

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Graphs encoding the generating properties of a finite group

Acciarri

Lucchini

2020

Mathematische Nachrichten

Self Cite

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“…Let G Sym(Ω) be a finite primitive permutation group with point stabiliser H = G α . Let us consider the relationship between d(G) and d(H We refer the reader to [16] for a recent application of Theorem 4.13 to the study of the exchange relation for generating sets of arbitrary finite groups.…”

Section: Primitive Permutation Groupsmentioning

confidence: 99%

Simple Groups, Generation and Probabilistic Methods

Burness¹

2019

Groups St Andrews 2017 in Birmingham

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It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years. In this survey article we discuss some of the extraordinary generation properties of simple groups, focussing on topics such as random generation, (a, b)-generation and spread, as well as highlighting the application of probabilistic methods in the proofs of many of the main results. We also present some recent work on the minimal generation of maximal and second maximal subgroups of simple groups, which has applications to the study of subgroup growth and the generation of primitive permutation groups.

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On the semigroup rank of a group

Branco

Gomes

2018

Semigroup Forum

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For an arbitrary group G, it is shown that either the semigroup rank Grk S equals the group rank Grk G , or Grk S = Grk G + 1. This is the starting point for the rest of the article, where the semigroup rank for diverse kinds of groups is analysed. The semigroup rank of relatively free groups, for any variety of groups, is computed. For a finitely generated abelian group G, it is proven that Grk S = Grk G + 1 if and only if G is torsion-free. In general, this is not true. Partial results are obtained in the nilpotent case. It is also proven that if M is a connected closed surface, then (π 1 (M ))rk S = (π 1 (M ))rk G + 1 if and only if M is orientable.

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Generating sets of finite groups

Cited by 11 publications

References 25 publications

Graphs encoding the generating properties of a finite group

Graphs encoding the generating properties of a finite group

Simple Groups, Generation and Probabilistic Methods

On the semigroup rank of a group

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