Half-automorphisms of free automorphic Moufang loops.
type: Item
instance of: scholarly article
title: Half-automorphisms of free automorphic Moufang loops.
zbMATH Open document ID: 1332.20068
DOI: 10.1134/S0001434615070342
author: N. E. Zubov
published in: Mathematical Notes
publication date: 2015-10-26T00:00:00Z
review text: In this note the authors study half-automorphisms of Moufang loops. They show that \textit{W. R. Scott}'s results [in Proc. Am. Math. Soc. 8, 1141-1144 (1958; Zbl 0080.24504)] hold for free automorphic Moufang loops. According to the authors for arbitrary automorphic Moufang loops, this is not known yet. The result of the authors is Theorem 3: Let \(A\) be a free automorphic Moufang loop, and let \(\tau\) be a half-automorphism on \(A\). Then \(\tau\) is an automorphism or \(\tau\) is an anti-automorphism of \(A\). -- Scott showed in the article mentioned above that any half-homomorphism of a group \(G\) into a group \(G'\) is either homomorphism or an anti-homomorphism. According to the authors in general, the theory of automorphism groups of Moufang loops is not very well developed. A recent paper of \textit{N. T. Lupashko} and \textit{N. I. Sandu} [Math. Notes 91, No. 3, 384-398 (2012); translation from Mat. Zametki 91, No. 3, 407-421 (2012; Zbl 1286.20077)] contains interesting results on automorphism groups of a certain class of infinite commutative Moufang loops. In that paper Lupashko and Sandu consider commutative Moufang loops \(Q\) with multiplication group \(M\) satisfying the minimality condition for its subloops. Such loops are characterized by various subgroups of automorphism groups. For this topic see also the paper of \textit{S. Saeki} [Trans. Am. Math. Soc. 346, No. 2, 707-718 (1994; Zbl 0837.43006)].
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20B27
zbMATH DE Number: 6498268
zbMATH Keywords: free automorphic Moufang loops
zbMATH Keywords: automorphic loops
zbMATH Keywords: automorphisms
zbMATH Keywords: automorphism groups
zbMATH Keywords: half-automorphisms
zbMATH Keywords: anti-automorphisms
reviewed by: C. Pereira da Silva
MaRDI profile type: Publication
cites work: Q3241211
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full work available at URL: s0001434615070342
OpenAlex ID: W2304195035
description: scientific article; zbMATH DE number 6498268