一些capable群

Some capable groups

对于一个给定的群G,如果能有另外的一个群H,使得H/Z(H)?G,则称群G是capable群,或称群G可以充当中心商.在p-群分类问题中对capable群进行的研究起着相当重要的作用.研究一些其所有的极大子群都同构且幂零类是2的capable群G,借助群的扩张理论,通过换位子计算,得到这类群是capable群的充要条件,并且由群G构造得到了群H,使得H满足H/Z(H)?G.

A group G was said to be capable if and only if G was isomorphic to H/Z(H) for some group H, where Z(H) was the center of H. The question of which p-groups were capable was interesting and played an important role in their classification. The author investigated the capable p-groups G which all maximal subgroup was isomorphism and nilpotent class was two. Using cyclic extension theory and commutator calculation, the paper determined some capable p-groups G and constructed H, such that G was isomorphic to H/Z(H).