A remark on socles and normal subgroups
type: Item
instance of: scholarly article
title: A remark on socles and normal subgroups
zbMATH Open document ID: 0539.20004
author: Tadashi Ikeda
published in: Hokkaido Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: Let k be an algebraically closed field and K a normal subgroup of the finite group G. Let W be an indecomposable kK-module, which is stable in G. It is shown that if W has a simple socle then so has any indecomposable direct summand of \(W^ G\) and the number of indecomposable summands of \(W^ G\) equals that of \((socle(W))^ G\). If W has a simple socle and if the characteristic of k is zero or prime to \(| G/K|\), then W is extendable to G iff socle(W) is. By passing to the contragredient modules one has the corresponding results for heads.
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 3857351
zbMATH Keywords: indecomposable modules
zbMATH Keywords: induced module
zbMATH Keywords: simple socle
zbMATH Keywords: indecomposable direct summand
zbMATH Keywords: heads
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381757714
OpenAlex ID: W2022695154
DOI: 10.14492/HOKMJ/1381757714
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description: scientific article; zbMATH DE number 3857351