Normal subgroups and projectivities of finite groups
type: Item
instance of: scholarly article
title: Normal subgroups and projectivities of finite groups
zbMATH Open document ID: 0581.20018
author: F. Menegazzo
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1979-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/107672
review text: Let G be a finite group, \(\sigma\) a projectivity of G onto \(G^{\sigma}\), H a normal subgroup of G and \(T^{\sigma}\) the core of \(H^{\sigma}\) in \(G^{\sigma}\). The author investigates the structure of H/T. He proves the following theorem: Let G be a finite p-group (p an odd prime), G/H cyclic, \(\sigma\) index-preserving and T trivial; then H is abelian. He remarks that no counterexamples to the case \(p=2\) are known. As a corollary of the main theorem, the author proves that if G is a group of odd order, then H/T is abelian. Examples of groups which satisfy the hypotheses of the theorem with \(H^{\sigma}\) nonabelian are given in the last section, using previous examples of groups with core- free quasinormal subgroups [\textit{J. G. Thompson}, Math. Z. 96, 226-227 (1967; Zbl 0189.317)].
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 3929262
zbMATH Keywords: projectivity
zbMATH Keywords: normal subgroup
zbMATH Keywords: core
zbMATH Keywords: finite \(p\)-group
zbMATH Keywords: core-free quasinormal subgroups
MaRDI profile type: Publication
cites work: \(p\)-subgroups of core-free quasinormal subgroups
cites work: Modulare Untergruppen endlicher Gruppen
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cites work: Q3232305
cites work: An example of core-free quasinormal subgroups of p-groups
description: scientific article; zbMATH DE number 3929262