弱c ##-正规子群及其性质

利用弱c ##-正规子群研究有限群的幂零性，得出以下结论：①设G是群， H ≤G ，若H在G中弱c ##-正规且H ≤M ≤G ，则H在M中弱c ##-正规.②设π为素数集，H是G的π-子群， N为G的正规π′-子群，如果H在G中弱c ##-正规，则HN/N在G/N中弱c ##-正规.③设G的每个素数阶元均为G的弱左Engle元，若2∈π（G），且G的每个4阶循环子群均在G中弱##c -正规，则G是幂零群.④设N〈G ， G/N为幂零的，且2∈π（G）.若N的每个素数阶元均为G的弱左Engle元，且N的每个4阶循环子群也在G中弱c##-正规，则G是幂零群.

On Weak c ##-Normality and Its Properties

The influence of Weakly ##c -normal Subgroup on the nilpotency of finite groups is investigated.①Suppose that G is a finite group, H ≤G , if H is weakly ##c -normal in G , and H ≤M ≤G , then H is weakly ##c -normal in M . ②Suppose that πis a prime set, H is a π-subgroup of G , N is a normalπ′-subgroup of G , if H is weakly ##c -normal in G , then HN/N is weakly ##c -normal in G/N .③Suppose that x is a weakly left Engle element for every element x of G with prime order, 2∈π（G） , if x is weakly ##c -normal in G for every element x of G with order 4, then G is nilpotent. ④ Suppose that N〈G , G/N is nilpotent, 2∈π（G） . If x is a weakly left Engle element for every element x of N with prime order, and x is weakly ##c -normal in G for every element x of N with order 4, then G is nilpotent.