有限群的弱c-正规子群及其性质

The Weakly c-normal Subgroups of Finite Groups and Their Properties

讨论了弱c-正规子群的性质,并利用其性质给出一个群为p-可解群、亚幂零群的一些条件.(1)设G为群,则G中存在弱c-正规Sylowp-子群当且仅当商群G/Op(G)为p-幂零群;特别地,G中存在弱c-正规Sylowp-子群时,G为p-可解群,且lp(G)≤2.(2)群G为亚幂零群当且仅当G的每一个Sylow子群在G中弱c-正规.

Discussion is made on the properties of weakly cnormal subgroups, and the conditions for psolvable group and metanilpotent group are determined as follows through the use of such properties: (1) Let G be a finite group, and then there exists the weakly cnormal Sylow psubgroup in G if and only if G/ Op(G) is pnilpotent; in particular, if there exists the weakly cnormal Sylow psubgroup in G, G is psolvable and lp(G) ≤2; (2) Let G be a finite group, and G is metanilpotent if and only if P is weakly cnormal for every Sylow subgroup P in G .