Relations between quandle extensions and group extensions

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.