Nonnormal and minimal nonabelian subgroups of a finite group.
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 another proof of a characterisation of Passman of non-Dedekind \(p\)-groups of order greater than \(p^3\) all of whose non-normal subgroups have the same order \(p\). In Section~3, a new proof of a theorem of Schmidt about groups with two classes of non-normal subgroups is presented. It is done with the help of a description of non-nilpotent groups \(G\) in which the number of non-\(G\)-invariant subgroups of pairwise distinct orders is~\(2\). Some gaps in the original proof of Schmidt are detected and his classification in the nilpotent case is corrected. The aim of Section~4 is to study groups with a unique minimal non-Abelian subgroup. Section~5 presents results about non-Dedekind \(p\)-groups all of whose non-normal subgroups of the same order are conjugate. It begins with the study with non-Dedekind \(p\)-groups in which all non-normal cyclic subgroups of the same order are conjugate. In Section~6, \(p\)-groups with exactly two conjugate classes of Abelian subgroups of type \((p,p)\), \(p>2\), are studied. The case of \(p\)-groups of order greater than \(p^p\) with exactly two conjugacy classes of subgroups of order \(p^{p-1}\) and exponent \(p\) is also studied. In Section~\(7\), \(p\)-groups with exactly two conjugacy classes of subgroups of a fixed order are considered. Section~8 presents results about \(p\)-groups with a small number of conjugacy classes of minimal non-Abelian subgroups. The paper concludes with a list of \(26\) open related problems.
scientific article; zbMATH DE number 5834349
scientific article