On connected transversals to dihedral subgroups
type: Item
instance of: scholarly article
zbMATH Open document ID: 0989.20057
author: Kari Myllylä
publication date: 2001-05-07T00:00:00Z
full work available at URL: isbn9514257669
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 1595697
zbMATH Keywords: solvable groups
zbMATH Keywords: solvable loops
zbMATH Keywords: solvability
zbMATH Keywords: connected transversals
zbMATH Keywords: inner mapping groups
MaRDI profile type: Publication
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title: On connected transversals to dihedral subgroups
published in: Acta Universitatis Ouluensis. Series A. Scientiae Rerum Naturalium
review text: Let \(G\) be a group with a dihedral subgroup \(H\) of order \(2x\), where \(x\) is an odd number. It is shown that, if there exist \(H\)-connected transversals \(A\) and \(B\) in \(G\), then \(G\) is a solvable group. (A subset \(A\) of \(G\) is said to be a left transversal to \(H\) if it contains exactly one element of each left coset of \(H\). If moreover \([A,B]<H\), then the transversals \(A\) and \(B\) are said to be \(H\)-connected.) This result applies to loop theory. It is proved that, if the inner mapping group \(I(Q)\) of a finite loop \(Q\) is dihedral of order \(2x\), then \(Q\) is a solvable loop.
reviewed by: A. M. Shelekhov
description: scientific article