Nonnormal and minimal nonabelian subgroups of a finite group.
type: Item
instance of: scholarly article
title: Nonnormal and minimal nonabelian subgroups of a finite group.
zbMATH Open document ID: 1216.20010
published in: Israel Journal of Mathematics
publication date: 2011-01-10T00:00:00Z
review text: Given a group theoretical property \(\mathfrak X\), we say that a group \(G\) is a minimal non-\(\mathfrak X\)-group if \(G\) does not satisfy \(\mathfrak X\), but all proper subgroups of \(G\) do. It seems clear that the study of minimal non-\(\mathfrak X\)-groups can give some insight into what makes a group satisfy the property \(\mathfrak X\). \textit{G. A. Miller} and \textit{H. C. Moreno} [Trans.\ Am.\ Math.\ Soc. 4, 398-404 (1903; JFM 34.0173.01)] began to study minimal non-Abelian groups. These groups were later classified by \textit{L. Rédei} [Comment.\ Math.\ Helv. 20, 225-267 (1947; Zbl 0035.01503)]. \textit{O. Schmidt} [Rec. Math. Moscou 31, 366-372 (1924; JFM 55.0672.04)] gave a description of minimal non-nilpotent groups. If the property \(\mathfrak X\) is closed under taking subgroups, a non-\(\mathfrak X\) should always contain a minimal non-\(\mathfrak X\)-group. This makes minimal non-\(\mathfrak X\)-groups a valuable technique in group theory. This paper is a good example of the application of minimal non-Abelian groups to present proofs of some deep theorems of the theory of finite groups. The author gives new proofs of some theorems of Janko, Schmidt, and Passman. In some cases, some gaps in the original results are corrected. He also proves new results, like some classification theorems. Since the paper contains a long list of theorems, this reviewer will give only a short description of the sections in which the paper is organised and will only summarise some of the theorems. Here \(p\) will mean a fixed prime. Section~1 is devoted to the study of non-Dedekind \(p\)-groups in which the non-normal cyclic subgroups have index~\(p\) (recall that a Dedekind group is a group with all subgroups normal). A new proof of Janko's characterisation of these groups is presented. In Section~2, groups with only one conjugacy class of non-normal subgroups are studied. Here a new proof of Schmidt's characterisation of these groups is presented. There is also a…
reviewed by: R. Esteban-Romero
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20E45
zbMATH DE Number: 5834349
zbMATH Keywords: minimal non-Abelian groups
zbMATH Keywords: minimal non-nilpotent groups
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: Dedekind groups
zbMATH Keywords: non-normal subgroups
zbMATH Keywords: non-Abelian subgroups
zbMATH Keywords: conjugacy classes of subgroups
author: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: s11856-010-0108-8
OpenAlex ID: W1992425600
cites work: On subgroups of finite \(p\)-groups
cites work: On subgroups and epimorphic images of finite \(p\)-groups
cites work: On Abelian subgroups of \(p\)-groups
cites work: Alternate proofs of some basic theorems of finite group theory
cites work: Some corollaries of Frobenius’ normal 𝑝-complement theorem
cites work: On the number of subgroups of given type in a finite p-group
cites work: Finite \(p\)-groups with few minimal nonabelian subgroups.
cites work: Groups of prime power order. Vol. 1.
cites work: Groups of prime power order. Vol. 2.
cites work: Q5480910
cites work: Q4220407
cites work: Generalizations of Certain Elementary Theorems on p -Groups
cites work: On a special class of \(p\)-groups
cites work: A complete classification of finite p-groups all of whose noncyclic subgroups are normal
cites work: Groups with few non-normal subgroups
cites work: A Contribution to the Theory of Groups of Prime-Power Order
cites work: Q4103728
cites work: On finite nonabelian 2-groups all of whose minimal nonabelian subgroups are of exponent 4.
cites work: A classification of finite 2-groups with exactly three involutions.
cites work: Finite 2-groups with no normal elementary Abelian subgroups of order 8
cites work: Q3371537
cites work: On finite groups with few non-normal subgroups
cites work: Nonnormal subgroups of \(p\)-groups.
cites work: The number of conjugacy classes of non-normal subgroups in nilpotent groups
cites work: Q2648167
cites work: Q4068906
cites work: Q5799445
DOI: 10.1007/S11856-010-0108-8
description: scientific article; zbMATH DE number 5834349