Large subgroups of small class in finite \(p\)-groups.
Abelian subgroups play a key role in the theory and applications of finite \(p\)-groups. For example, J. G. Thompson strengthened his celebrated theorem, the Frobenius kernel of a Frobenius group is nilpotent, by using the Abelian subgroups of maximal rank in a Sylow subgroup of a finite group [\textit{J. G. Thompson}, J. Algebra 1, 43-46 (1964; Zbl 0119.26802)]. One may ask whether one of these subgroups is normal in the Sylow subgroup. More generally, the following very interesting questions are posed in the present paper: Question 1. Suppose \(A\) is an elementary Abelian subgroup of order \(p^n\) in a finite \(p\)-group \(P\). Does \(P\) possess a normal elementary Abelian subgroup of order \(p^n\)? Question 2. Suppose \(A\) is an Abelian subgroup of order \(p^n\) in a finite \(p\)-group \(P\). Does \(P\) possess a normal Abelian subgroup of order \(p^n\)? In general, the answers to the above questions are all ``no'', but affirmative under certain conditions. In a remarkable paper [\textit{J. L. Alperin} and \textit{G. Glauberman}, J. Algebra 203, No. 2, 533-566 (1998; Zbl 0964.20011)], using an idea from algebraic group theory, the authors provided affirmative answers to some special cases of Question 1 and obtained some related results for \(p\)-groups and nilpotent Lie algebras over fields of characteristic \(p\) under the restriction on \(A\) being elementary Abelian. In this paper, the author removes the restriction and obtains analogues for Question 2 of the results about Question 1 by extending the application of algebraic group theory. He also obtains generalizations to subgroups of small nilpotence class, not necessarily Abelian, and to nilpotent Lie algebras over local rings. Five rather complicated theorems are proved in the present paper.
scientific article; zbMATH DE number 2057667
scientific article