Multiplication groups of commutative automorphic \(p\)-loops of odd order are \(p\)-groups.
type: Item
instance of: scholarly article
title: Multiplication groups of commutative automorphic \(p\)-loops of odd order are \(p\)-groups.
zbMATH Open document ID: 1250.20059
author: Piroska Csörgő
published in: Journal of Algebra
publication date: 2012-05-23T00:00:00Z
review text: The study of automorphic loops (A-loops) was initiated by \textit{R. H. Bruck} and \textit{L. J. Paige}, [Ann. Math. (2) 63, 308-323 (1956; Zbl 0074.01701)]. In this paper the author proves without using the nilpotence that the multiplication group of a finite commutative A-loop of order a power of an odd prime \(p\) is a \(p\)-group. Consequently the loop has nontrivial center which implies the central nilpotence of the loop. The author proves this result by applying the theory of connected transversals that was introduced by \textit{M. Niemenmaa} and \textit{T. Kepka} [in J. Algebra 135, No. 1, 112-122 (1990; Zbl 0706.20046)].
reviewed by: C. Pereira da Silva
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 6037589
zbMATH Keywords: A-loops
zbMATH Keywords: multiplication groups
zbMATH Keywords: inner mapping groups
zbMATH Keywords: center
zbMATH Keywords: central nilpotence
zbMATH Keywords: translations
zbMATH Keywords: connected transversals
zbMATH Keywords: commutative automorphic loops
zbMATH Keywords: finite loops
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.09.038
OpenAlex ID: W2077150039
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DOI: 10.1016/J.JALGEBRA.2011.09.038
description: scientific article; zbMATH DE number 6037589