Sufficient conditions for the solvability and supersolvability in finite groups
type: Item
instance of: scholarly article
title: Sufficient conditions for the solvability and supersolvability in finite groups
zbMATH Open document ID: 0544.20019
DOI: 10.1016/0022-4049(84)90057-4
author: M. Asaad
published in: Journal of Pure and Applied Algebra
publication date: 1984-01-01T00:00:00Z
review text: A finite group G is called an H-r N-group if i) G has even order and ii) each even-ordered subgroup H of G with \(| H|\) the product of r not necessarily distinct primes, is normal in G. The author proves the following results: 1. If G is an H-2 N-group that does not involve \(A_ 4\), then G is supersolvable. 2. If G is an H-2 N-group or an H-3 N-group then G' is nilpotent. 3. If G is an H-4 N-group then either G is solvable or G is isomorphic with \(L_ 2(q)\) for certain odd prime powers q. The author also shows that if G is an odd order group in which each subgroup of order pq is normal (where p and q are not necessarily distinct primes) then G' is nilpotent. Finally, the author proves that a group G in which each subgroup of order \(p^ 2\) is normal, for every prime factor of \(| G|\) except perhaps the largest, is either Sylow-towered in the natural order or involves \(A_ 4\). The work is related to a paper by \textit{N. S. Sastry} and the reviewer on the influence of normality conditions on almost minimal subgroups [J. Algebra 52, 364-377 (1978; reviewed above)].
reviewed by: W. E. Deskins
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D25
zbMATH DE Number: 3865557
zbMATH Keywords: H-r N-group
zbMATH Keywords: \(L_ 2(q)\)
zbMATH Keywords: normality conditions
zbMATH Keywords: almost minimal subgroups
MaRDI profile type: Publication
full work available at URL: 0022-4049(84)90057-4
OpenAlex ID: W2067923883
cites work: Influence of normality conditions on almost minimal subgroups of a finite group
cites work: Q5577154
cites work: Q3240869
cites work: Endliche Gruppen I
cites work: Q5331572
cites work: On minimal subgroups which are normal.
cites work: Q3244127
cites work: Solvability of groups of odd order
cites work: Subgroups of finite groups
cites work: Finite groups with invariant fourth maximal subgroups
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description: scientific article; zbMATH DE number 3865557