子群指数给出的超可解群的一个充要条件

A Necessary and Sufficient Condition of Supersolvable Group Offered by Index of Subgroup

首先证明了如果群G有一个正规循环子群H,使对应的商群G/H是超可解的,则G本身必定是超可解群;然后证明了G的子群H有一切可能指数的子群,当且仅当它有一切可能素指数的子群,最后证明了群G为超可解当且仅当它的每个子群含有指数为任意素数的子群.

This paper proved first that group G itself must be supersolvable group if G has a normal cyclic subgroup H and the corresponding quotient group G/H should be supersolvable; Then, proved that subgroup H of G has all subgroups with possible index if it has all subgroups with possible prime index; Finally, proved that G should be supersolvable if every subgroup has the index with arbitrary prime number.