Relations between quandle extensions and group extensions

Yongju Bae, J. Scott Carter, Byeorhi Kim

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

In [6] and [7], Joyce and Matveev showed that for given a group G and an automorphism ϕ, there is a quandle structure on the underlying set of G. When the automorphism is an inner-automorphism by ζ, we denote this quandle structure as (G,◃ζ). In this paper, we show a relationship between group extensions of a group G and quandle extensions of the quandle (G,◃ζ). In fact, there exists a group homomorphism from Hgp2(G;A) to Hq2((G,◃ζ);A). Next, we show a relationship between quandle extensions of a quandle Q and quandle extensions of the quandle on the inner automorphism group of Q. Indeed, there exists a group homomorphism from Hq2(Q;A) to Hq2((Inn(Q),◃ζ);A). Finally, we observe via examples a relationship between extensions of a quandle and extensions of the inner automorphism group of the quandle.

Original languageEnglish
Pages (from-to)410-435
Number of pages26
JournalJournal of Algebra
Volume573
DOIs
StatePublished - 1 May 2021

Keywords

  • Abelian extension
  • Central extension
  • Group 2-cocycle
  • Quandle 2-cocycle

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