关于有限群的正规子群的补子群Ⅱ

研讨了关于有限群G的一个正规子群K的补子群之存在性与共轭性的更多一些的结果。主要结果如下：(1)假设K是Abel群并且K的每个Sylow子群S在G之含S的Sylow子群中有补子群，则有：(i)K在G中有补子群；(ii)若G有Hall π—子群H，其中π=π(K)，并且K在H中的所有补子群在H中是共轭的，则K在G中的所有补子群在G中是共轭的，(2)假设K是可解的并且对所有的S／K∈Syl(G／K)，K是S的一个直因子，则有：(i)K在G中有补子群；(ii)若G有Hall π—子群H，其中π=π(K)，则K在G中的所有补子群在G中共轭的充要条件是K在H中的所有补子群在H中共轭。

On Complements of Normal Subgroups in Finite Groups Ⅱ

In this paper, some more properties of the existence and conjugacy of complements of a normal subgroup K of a finite group G are studied. The main results are as follows. (1) Suppose that K is abelian and every Sylow subgrop S of K has a complement in a Sylow subgroup of G which contains S. Then: (i) K has a complement in G; (ii) If G has a Hall π- subgroup H with π = π(K), and all complements of K in H are conjugate in H, then all complements of K in G are conjugate in G. (2) Suppose that K is solvable and K is a direct factor of S for each S/K∈ Syl(G/K).Then: (i) K has a complement in G;(ii) If G has a Hall π-subgroup H with π = π(K), then all complements of K inG are conjugate in G ff and only if all complements of K in H are conjugate in H.