On the existence of Hall subgroups
This paper is a happy reminder of how beautiful finite group theory can still be. It is concerned with theorems like Sylow's and the conditions \(E_{\pi}\), \(C_{\pi}\), \(D_{\pi}\) introduced by Hall; the group G satisfies \(E_{\pi}\) if G possesses Hall \(\pi\)-subgroups, \(C_{\pi}\) if G has a unique class of them and \(D_{\pi}\) if there is a Hall \(\pi\)- subgroup of G which contains a conjugate of any \(\pi\)-subgroup of G. Assuming the Schreier hypothesis, (as is done throughout the paper), Chunikhin showed that if N is normal in G and N, G/N both satisfy \(C_{\pi}\), then G satisfies \(C_{\pi}.\) But the corresponding statement for \(E_{\pi}\) is false. It is proved (Theorem 3.4) that if N and G/N both satisfy \(E_{\pi}\) but G does not, then G has a non-Abelian composition factor H/K such that H/K does satisfy \(E_{\pi}\) but \(Aut_ G(H/K)\) does not. Here \(Aut_ G(H/K)\) means the group of automorphisms of H/K induced by \(N_ G(H)\cap N_ G(K)\), so it is dependent on H and K, not merely on the composition factor H/K. In fact an example is given of a group G with two composition series, where all the factors X of one fulfil the condition that \(Aut_ GX\) satisfies \(E_{\pi}\) but some of those of the other do not. Let us start, then, with the condition (d) that \(Aut_ G(H/K)\) satisfies \(E_{\pi}\) for all composition factors H/K of all composition series of G. Let \(1=G_ 0<G_ 1<...<G_ n=G\) be a composition series which is a refinement of a chief series of G. Trivially, (d) implies the condition (c) that \(Aut_ G(G_ i/G_{i-1})\) satisfies \(E_{\pi}\) for all i. But also such a series has the property that if A is an atom of \(G_ i\) with maximal normal subgroup \(A^*\), then \(A/A^*\cong G_ i/G_{i- 1}\) and \(Aut_ G(A/A^*)\cong Aut_ G(G_ i/G_{i-1})\) (Lemma 2.5). (An atom of G, as defined by Wielandt, is a perfect subnormal subgroup with a unique maximal normal subgroup.) So (c) implies the condition (b) that \(Aut_ G(A/A^*)\) satisfies \(E_{\pi}\) for every atom A of G. In fact (b) implies (d), as is shown using a theorem of Wielandt. The main theorem (Theorem 3.5) asserts that these three conditions (b), (c), (d) are equivalent to the condition (a) that H satisfies \(E_{\pi}\) whenever H is a subgroup of G for which the terminal member of the derived series is subnormal in G. The fact that (a) implies (b) is fairly straightforward, but the converse uses Theorem 3.4 stated above. The final section of the paper does the same sort of thing for \(D_{\pi}\). Theorem 4.5 states that if M is a minimal normal subgroup of G satisfying \(C_{\pi}\) and G/M satisfies \(D_{\pi}\) but G does not, then M contains a normal simple non-Abelian subgroup S such that S satisfies \(C_{\pi}\) but \(Aut_ G(S)\) does not satisfy \(D_{\pi}\). And Theorem 4.6 shows among other things that H satisfies \(D_{\pi}\) for any subgroup H of G for which the terminal member of the derived series is subnormal in G if and only if \(Aut_ G(H/K)\) satisfies \(D_{\pi}\) and H/K satisfies \(C_{\pi}\) for all composition factors H/K of G.
scientific article; zbMATH DE number 3944134
scientific article