On multiplication groups of left conjugacy closed loops. (English) Zbl 1101.20035

Drápal, Aleš

Commentat. Math. Univ. Carol. 45, No. 2, 223-236 (2004).

Summary: A loop \(Q\) is said to be left conjugacy closed (LCC) if the set \(\{L_x;\;x\in Q\}\) is closed under conjugation. Let \(Q\) be such a loop, let \(\mathcal L\) and \(\mathcal R\) be the left and right multiplication groups of \(Q\), respectively, and let \(\text{Inn\,}Q\) be its inner mapping group. Then there exists a homomorphism \({\mathcal L}\to\text{Inn\,}Q\) determined by \(L_x\mapsto R^{-1}_xL_x\), and the orbits of \([\mathcal{L,R}]\) coincide with the cosets of \(A(Q)\), the associator subloop of \(Q\). All LCC loops of prime order are Abelian groups.

MSC:

20N05

Loops, quasigroups

Keywords:

left conjugacy closed loops; multiplication groups; nuclei; inner mapping groups