On Sylowizers in finite \(p\)-soluble groups.
In a finite group \(G\) the Sylowizer of a \(p\)-subgroup \(R\) is a subgroup maximal with respect to having \(R\) a Sylow \(p\)-subgroup. This concept is due to \textit{W. Gaschütz} [Math. Z. 122, 319-320 (1971; Zbl 0234.20008)], where the question of conjugacy of the Sylowizers for \(p\)-solvable groups \(G\) was studied. \textit{A. V. Borovik} and \textit{E. I. Khukhro} [Math. Zametki 19, 401-418 (1976; Zbl 0347.20011)] found examples of Sylowizers of different orders. In this paper sufficient conditions are given for the conjugacy of the Sylowizers, such as, (1) \(p=2\) and the \(2\)-Sylow subgroups of the group \(G\) are either dihedral or quaternion; (2) the \(p\)-Sylow subgroups of the group \(G\) are of order at most \(p^3\); (3) \(p\) is an odd prime, the subgroup \(R\) is Abelian of maximal order.
scientific article; zbMATH DE number 5292353