幂零群的若干充分条件

在文献1]的基础上，改变-些条件得出G为幂零群的若干充分条件。利用弱C-正规，s-正规与弱左Engle元之间的关系获得了下面几个定理：①G的每个素数阶元均为G的弱左Engle元；如果2∈φ（G），G的每个4阶循环子群均在G中弱C-正规，则G是幂零群。②设N〈3G，G／N幂零，2∈π（G），若N的素数阶元均为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幂零。③如果G的每个素数阶元x为NG（（x））的弱左Engle元，并且〈x〉和G的每个4阶循环子群均在G中弱C-正规，则G是幂零群。④G的每个素数阶元均为G的弱左Engle元；如果2∈π（G），G的每个4阶循环子群均在G中S-正规，则G是幂零群。⑤如果G的每个素数阶元x为NG（（x））的弱左Engle元，并且（x）和G的每个4阶循环子群均在G中弱S-正规，则G是幂零群。

Some Sufficient Conditions of a Niipotent Group

The author discusses this topic and obtain some theorems by using the relation between weakly C-normality of subgroups and weakly left Engle element in a finite group. （1）suppose that x is a weak left Engle element for every element x of G with prime order, 2 ∈ φ（ G）, if 〈x） is weak C-normal in G for every element x of G with order 4. Then G is nilpotent. （2）suppose that N△G, G/N G is nilpotent, 2 ∈ π（G）. If x is a weak left Engle element for every element x of N with prime order, and 〈x〉 is weak C-normal in G for every element x of N with order 4. Then G is nilpotent. （3）Let G be a finite group. Suppose that every element x of G with a prime order p is a weakly left Engle element of, all cyclic subgroups of G with order 4 and 〈x〉 are weakly C-normal in G. Then G is nilpotent. （4） suppose tbat x is a weak left Engle element for every element x of G with prime order, 2 ∈ φ（G）, if 〈x〉 is weak S-normal in G for every element x of G with order 4. Then G is nilpotent. （5）Let G be a finite group. Suppose that every element x of G with a prime order p is a weakly left Engle element of G, all cyclic subgroups of G with order 4 and 〈x〉 are weakly S-normal in G. Then G is nilpotent.