Normalizer quotients of symmetric groups and inner holomorphs

Alexei Entin*, Cindy (Sin Yi) Tsang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Article number107839
JournalJournal of Pure and Applied Algebra
Volume229
Issue number1
DOIs
StatePublished - Jan 2025

Funding

FundersFunder number
Israel Science Foundation2507/19

    Keywords

    • Finite group
    • Inner holomorph
    • Normalizer quotient
    • Symmetric group

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