关于有限群的正规子群的补子群I

研讨了一个有限群的正规子群的补子群之存在性与共轭性的若干结果，主要的结果如下：设G／K是π-可解的并设日为有限群G的一个Hallπ-子群，其中π=π(K)，则有：(1)若K的每个Syylow子群Pl在G的某个含P1的Sylow子群中有补子群并且这个补子群在G中半正规，则K在G中有补子群，(2)若进一步设K在H中的所有补子群(由(1)，这些补子群存在，)在H中共轭，则K在G中的所有补子群在G中共轭。

On Complements of Normal Subgroups in Finite Groups I

In this paper, some resutls on the existence and conjugacy of complements of a normal subgroup of a finite group are studied. The main result is as follows. Let G/K be π-solvable and let H be a Hall π-subgroup of Afinite group G with π=π(K), (1) if every Sylow subgroup P1 of K has a complement in a Sylow subgroup P of G containing P1 and all complements of P1 in P are seminormal in G, then K has a complement in G and (2) if, moreover, all complements of K in H are conjugate in H, then all complements of K in G are conjugate in G.