有限p—幂零群的一个新刻划

推广了Itδ的结果,得到下述主要定理.定理1 设G是有限群,N(?)G,G/N p-幂零.那么(i)p为奇素数时,G p-幂零当且仅当N的p阶元均含于Z_(p∞)(G);(ii)p=2时,G 2-幂零当且仅当N的2.2~2阶元均含于Z_(2∞)(G).定理2 设G是有限群,N(?)G且G/N是幂零群.那么G是幂零群当且仅当N的素数阶元与2~2阶元均.含于Z_∞(G).此外,还证明了定理3 设G是有限群.则Z_(p∞)(G)=NI_(G)=∩{M|M为G的极大p-幂零子群}.

A NEW DESCRIPTION OF FINITE p -NILPOTENT GROUPS

This paper extends ltd result completely, obtains the following main theoremsTherein 1 Let G be a finite group, N G. G/N p -nilpotent.(i) If p is a odd prime, then G is p -nilpotent if and only if all elements of p order of N belong to Z,oo (G).(ii) If p = 2 , then G is 2-nilpotent if and only if all elements of 2, 22 order of N belong to Z200 (G) .Theorem 2 Let G be a finite Group, N Gand G/N is nilpotent. Then G is nilpotent If and only if all elements of prime and 2J order of N belong to Z,(G) .In addition, it provesTheorem 3 Let G be a finite Group. ThenZ (G) =NI,(G) = (M \M is a maximal p -nilpotent subgroup of G }