On abelian-by-cyclic Moufang loops

2023/01/09 by Drápal, Aleš, Vojtěchovský, Petr
#20N05 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2301.03683

Abstract

We study abelian-by-cyclic Moufang loops. We construct all split 3-divisible abelian-by-cyclic Moufang loops from so-called Moufang permutations on abelian groups (X,+), which are permutations that deviate from an automorphism of (X,+) by an alternating biadditive mapping (satisfying certain properties). More generally, we obtain additional abelian-by-cyclic Moufang loops from so-called construction pairs. As an aside, we show that in the Moufang loops Q obtained from a construction pair on (X,+) the abelian normal subgroup (X,+) induces an abelian congruence of Q if and only if Q is a group.

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• Congruence solvability in finite Moufang loops of order coprime to three
• Cyclic extensions of Moufang loops induced by semi-automorphisms
• Moufang semidirect products of loops with groups and inverse property extensions
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