Strict inequalities for minimal degrees of direct products

Neil J Saunders

doi:10.1017/S0004972708000956

Strict inequalities for minimal degrees of direct products

Neil J Saunders

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Abstract

The minimal faithful permutation degree μ(G) of a finite group G is the least non-negative integer n such that G embeds in the symmetric group Sym(n). Work of Johnson and Wright in the 1970s established conditions for when μ(H×K)=μ(H)+μ(K), for finite groups H and K. Wright asked whether this is true for all finite groups. A counter-example of degree 15 was provided by the referee and was added as an addendum in Wright’s paper. Here we provide two counter-examples; one of degree 12 and the other of degree 10.

Original language	English
Pages (from-to)	23-30
Number of pages	8
Journal	Bulletin of the Australian Mathematical Society
Volume	79
Issue number	1
DOIs	https://doi.org/10.1017/S0004972708000956
Publication status	Published - Feb 2009

Keywords

faithful permutation representations; complex reflection groups; monomial reflection groups

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title = "Strict inequalities for minimal degrees of direct products",

abstract = "The minimal faithful permutation degree μ(G) of a finite group G is the least non-negative integer n such that G embeds in the symmetric group Sym(n). Work of Johnson and Wright in the 1970s established conditions for when μ(H×K)=μ(H)+μ(K), for finite groups H and K. Wright asked whether this is true for all finite groups. A counter-example of degree 15 was provided by the referee and was added as an addendum in Wright{\textquoteright}s paper. Here we provide two counter-examples; one of degree 12 and the other of degree 10.",

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AB - The minimal faithful permutation degree μ(G) of a finite group G is the least non-negative integer n such that G embeds in the symmetric group Sym(n). Work of Johnson and Wright in the 1970s established conditions for when μ(H×K)=μ(H)+μ(K), for finite groups H and K. Wright asked whether this is true for all finite groups. A counter-example of degree 15 was provided by the referee and was added as an addendum in Wright’s paper. Here we provide two counter-examples; one of degree 12 and the other of degree 10.

KW - faithful permutation representations; complex reflection groups; monomial reflection groups

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DO - 10.1017/S0004972708000956

M3 - Article (Academic Journal)

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VL - 79

SP - 23

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JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

IS - 1

ER -

Strict inequalities for minimal degrees of direct products

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