Conjugacy closed loops and their multiplication groups. (English) Zbl 1047.20049

Drápal, Aleš

J. Algebra 272, No. 2, 838-850 (2004).

Author’s abstract: A loop is said to be conjugacy closed if the sets \(\{L_x:x\in Q\}\) and \(\{R_x:x\in Q\}\) are closed under conjugation. Let \(L\) and \(R\) be the left and right multiplication groups of \(Q\) respectively, and let \(I(Q)\) be its inner mapping group. If \(Q\) is conjugacy closed then there exist epimorphisms \(L\to I(Q)\) and \(R\to I(Q)\) that are termined by \(L_x\mapsto R^{-1}_xL_x\) and \(R_x\mapsto L^{-1}_xR_x\). These epimorphisms are used to expose various structural properties of \(M(Q)=LR\).

Reviewer: Miklos Csikós (Gödöllö)

MSC:

20N05

Loops, quasigroups

Keywords:

conjugacy closed loops; multiplication groups; nuclei; translations

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