On the groups generated by a class of \(p\)-involutions.
type: Item
instance of: scholarly article
title: On the groups generated by a class of \(p\)-involutions.
zbMATH Open document ID: 0685.20016
DOI: 10.1007/BF01190075
published in: Archiv der Mathematik
publication date: 1991-01-01T00:00:00Z
review text: A group of order \(2p^ n\) (\(p\) prime) is said to be an \(F(p)\)-group if it is spanned by its involutions. These groups have been considered by \textit{B. Fischer} [Math. Z. 83, 267-303 (1964; Zbl 0119.26201)], and (for \(p=3)\) by \textit{L. Bénéteau} [C. R. Acad. Sci., Paris, Sér. A 285, 735-737 (1977; Zbl 0389.20026)]. Let \(P\) be a \(p\)-group. It is shown that there is an \(F(p)\)-group \(G\) such that \(D(G)=P\) if and only if \(P\) has an automorphism \(f\) such that \(P=\langle x\mid f(x)=x^{-1}\rangle\). Whenever it exists, this \(G\) is unique up to isomorphism. If \(P\) has an automorphism \(f\) as above, this holds also for \(P\wr C_ q\) (\(q\) a power of \(p\)). Finally, every finite \(p\)-group can be embedded in an \(F(p)\)-group.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 4122210
zbMATH Keywords: involutions
zbMATH Keywords: automorphisms
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: \(F(p)\)-groups
author: Raffaele Scapellato
reviewed by: Raffaele Scapellato
MaRDI profile type: Publication
cites work: Q4135773
cites work: Q4170897
cites work: Distributive Quasigruppen endlicher Ordnung
cites work: On p-groups with odd order automorphism groups
cites work: Q4402061
cites work: Q5569508
cites work: Q4146884
full work available at URL: https://doi.org/10.1007/bf01190075
OpenAlex ID: W2095472387
description: scientific article; zbMATH DE number 4122210