非正规子群都是q群的有限群

Non-normal Subgroup Are Finite Groups of q-Groups

得到非正规子群都是q群的完全分类,即证明了如下结论:设q是一个素数,有限群G不是Dedekind群,则G的非正规子群都是q群的充要条件是G为非交换q群且不同构于Q8×E,其中Q8是8阶四元数群,E为初等阿贝尔2-群,或G=PQ,其中P为G的p阶正规子群,Q为G的非正规q群,Q为Dedekind群且p=1(modq).

In this paper we had discussed that all non-normal subgroups are finite groups of q-groups. And we proved the following theorem：Let q be a prime,finite group G is not a Dedekind group. Then all non-normal subgroups if G are q-groups if and only if G is a non abelian q-group and G is not ismorphic to Q8× E,where Q8 is a quaternion group of order 8,E is an elementary 2 - group or G = PQ,where P is a normal subgroup of G with order p,Q is a non-normal Sylow q-subgroup of G,Q is a Dedekind group and q=1（mod p）.