有限群的p-幂零性的一个注记

On p-Nilpotence of Finite Groups

推广了c-补,并给出有限群p-幂零性的一个新判别条件.设G是一个有限群,H是G的一个子群.如果存在G的一个子群K,使得G=HK,且H∩K≤H■,这里H■表示G的包含在H中的最大拟正规子群,则称K是H在G中的一个弱c-补,H在G中有一个弱c-补.证明了:设p是G的阶的最小素因子,P是G的一个Sylowp-子群,若P的每个2-极大子群在G中有弱c-补,且G与A4无涉,则G是p-幂零的.

c-Supplement is generalized and weak c-supplement is introduced. Let G be a finite group and H a subgroup of G. H is called weakly c-supplemented in G if there exists a subgroup K of G such that G = HK and H∩K≤Hg, where Hg is the maximal quasinormal subgroup of G contained in H. K is called a weak c-supplement of H in G. It is shown that for the least prime divisor p of the order of a finite group G and for a Sylow p-subgroup P of G, if every 2-maximal subgroup of P is c-supplemented in G and if G is not involved with A4, then G is p-nilpotent.