关于强半正规性

ON STRONG SEMI-NORMALITY

引入了关于有限群的子群的一个新概念:H≤G,H的每Sylow子群在G内半正规,则称H在G内强-半正规.利用这一概念,我们证明了如下的结果:(1)群G中,强-半正规的极大子群是超可解的,且在G中有素数指数.(2)存在强-半正规极大子群的群是可解群.(3)若群G的极大子群M强-半正规,且M的每Sylow子群的极大子群在G内半正规。

A new concept on subgroups of finite groups was introduced: a subgroup H of a finite group G is called to be strong semi-normal if H<G and any Sylow. subgroups of H arc semi-normal in G. The following results were proven with the concept.(l) Let M be a maximal subgroup of a finite group G and if it is strong semi-normal in G, then M is supersolvable and the index of M in G is a prime number.(2) A finite group G is solvable, if there exists a maximal subgroup which is strong semi-normal in G.(3) A finite group G is supersolvable, if there exists a maximal subgroup M which is strong semi-normal and the maximal subgroups of any Sylow subgroups of M are semi-normal in G.