Normalizer quotients of symmetric groups and inner holomorphs

Alexei Entin; Cindy (Sin Yi) Tsang

doi:10.1016/j.jpaa.2024.107839

Normalizer quotients of symmetric groups and inner holomorphs

Alexei Entin, Cindy (Sin Yi) Tsang

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

We show that every finite group T is isomorphic to a normalizer quotient N_{S_n}(H)/H for some n and a subgroup H≤S_n. We show that this holds for all large enough n≥n₀(T) and also with S_n replaced by A_n. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group G with Out(G)≅T (for any given finite group T) and the determination of the normalizer in Sym(G) of the inner holomorph InHol(G)≤Sym(G) for any centerless indecomposable finite group G, which may be of independent interest.

Original language	English
Article number	107839
Journal	Journal of Pure and Applied Algebra
Volume	229
Issue number	1
DOIs	https://doi.org/10.1016/j.jpaa.2024.107839
State	Published - Jan 2025

Keywords

Finite group
Inner holomorph
Normalizer quotient
Symmetric group

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

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10.1016/j.jpaa.2024.107839

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title = "Normalizer quotients of symmetric groups and inner holomorphs",

abstract = "We show that every finite group T is isomorphic to a normalizer quotient NSn(H)/H for some n and a subgroup H≤Sn. We show that this holds for all large enough n≥n0(T) and also with Sn replaced by An. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group G with Out(G)≅T (for any given finite group T) and the determination of the normalizer in Sym(G) of the inner holomorph InHol(G)≤Sym(G) for any centerless indecomposable finite group G, which may be of independent interest.",

keywords = "Finite group, Inner holomorph, Normalizer quotient, Symmetric group",

author = "Alexei Entin and Tsang, {Cindy (Sin Yi)}",

note = "Publisher Copyright: {\textcopyright} 2024 Elsevier B.V.",

year = "2025",

month = jan,

doi = "10.1016/j.jpaa.2024.107839",

language = "الإنجليزيّة",

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journal = "Journal of Pure and Applied Algebra",

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T1 - Normalizer quotients of symmetric groups and inner holomorphs

AU - Entin, Alexei

AU - Tsang, Cindy (Sin Yi)

PY - 2025/1

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N2 - We show that every finite group T is isomorphic to a normalizer quotient NSn(H)/H for some n and a subgroup H≤Sn. We show that this holds for all large enough n≥n0(T) and also with Sn replaced by An. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group G with Out(G)≅T (for any given finite group T) and the determination of the normalizer in Sym(G) of the inner holomorph InHol(G)≤Sym(G) for any centerless indecomposable finite group G, which may be of independent interest.

AB - We show that every finite group T is isomorphic to a normalizer quotient NSn(H)/H for some n and a subgroup H≤Sn. We show that this holds for all large enough n≥n0(T) and also with Sn replaced by An. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group G with Out(G)≅T (for any given finite group T) and the determination of the normalizer in Sym(G) of the inner holomorph InHol(G)≤Sym(G) for any centerless indecomposable finite group G, which may be of independent interest.

KW - Finite group

KW - Inner holomorph

KW - Normalizer quotient

KW - Symmetric group

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U2 - 10.1016/j.jpaa.2024.107839

DO - 10.1016/j.jpaa.2024.107839

M3 - مقالة

SN - 0022-4049

VL - 229

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 1

M1 - 107839

ER -

Normalizer quotients of symmetric groups and inner holomorphs

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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