On the Hall \(D_ \pi\)-property for finite groups
A finite group is said to satisfy the property \(E_\pi\), if it possesses a Hall \(\pi\)-subgroup; \(C_\pi\), if it satisfies the property \(E_\pi\) and all its Hall \(\pi\)-subgroups are conjugate; \(D_\pi\), if it satisfies the property \(C_\pi\) and every \(\pi\)-subgroup of it is contained in some Hall \(\pi\)-subgroup. The above notation was introduced by P. Hall and the properties generalize those of a Sylow \(p\)-subgroup. P. Hall also raised the following question: Is an extension of a \(D_\pi\)-group by another \(D_\pi\)-group again a \(D_\pi\)-group? There is no exhaustive answer to this question, although an affirmative answer for \(C_\pi\)-groups is well known. Let \(S\) be a normal and homomorphism-closed class of finite groups. Say that the \(D_\pi\)-conjecture is valid for \(S\) if an extension of every \(D_\pi\)-group in \(S\) by an arbitrary \(D_\pi\)-group is a \(D_\pi\)-group. Say that the \(D_\pi\)-conjecture is valid for almost every simple group in \(S\) if, for every nonabelian simple group \(P\in S\), the group \(\text{Out}(P)\) is solvable and the property \(D_\pi\) is satisfied for every group \(G\) that satisfies the following conditions: 1) \(\text{Inn}(P)\leq G\leq\Aut(P)\), with \(P\) a nonabelian simple group; 2) \(G\) possesses a normal subgroup \(A\) in \(S\) that contains \(P\) and satisfies the property \(D_\pi\). Call a class \(S\) \(D_\pi\)-normal if the property \(D_\pi\) is hereditary for normal subgroups in \(S\). The authors of the present article show that the general Hall problem can be reduced to an examination of simple groups and their automorphism groups. In particular, they prove the following result: Theorem 1. Let \(S\) be a normal and homomorphism-closed class of finite groups. If the \(D_\pi\)-conjecture is valid for almost every simple group in \(S\) then it is valid for \(S\). Theorem 2 provides sufficient conditions for \(D_\pi\)-normality. These two theorems arouse the hope that an answer to P. Hall's question can be given with the help of the classification of finite simple groups. Furthermore, they allow the authors to distinguish two new classes of groups satisfying the \(D_\pi\)-conjecture: 1) the class of groups all whose composition factors are isomorphic to the alternating groups or have prime order (Theorem 3), and 2) the class of groups whose composition factors possess Abelian Sylow 2-subgroups (Theorem 4). Notice that, earlier, Shemetkov found some classes satisfying the \(D_\pi\)-conjecture (for example, see \textit{L. A. Shemetkov} [Mat. Sb., Nov. Ser. 76(118), 271-287 (1968; Zbl 0212.04902)]).
scientific article; zbMATH DE number 1099651