具有X-s-半置换的准素子群的有限群

Finite groups with X-s-semipermutable primary subgroups

设X是群G的非空子集,H是G的子群,如果H在G中有一个补充T使得H和T的所有Sylow子群X-置换,则称H在G中X-s-半置换.利用子群的X-s-半置换性得到下列结果：①设F是包含所有超可解群的饱和群系,X是群G的可解正规子群,则G∈F当且仅当存在H G使得G/H∈F且H的每个Sylow子群的每个极大子群在G中X-s-半置换.②设F是包含所有超可解群的饱和群系,X是群G的可解正规子群且H G.如果G/H∈F且~F（H）的每个Sylow子群的每个极大子群在G中X-s-半置换,则G∈F.③设X是群G的一个p-可解正规子群,p是｜G｜的最小素因子.如果G是A4-自由的,且存在H G使得G/H是p-幂零的并满足H的每个Sylowp-子群的每个2-极大子群在G中X-s-半置换,那么G是p-幂零的.

Let X be a nonempty subset of a group G. A subgroup H of G is said to be X-s-semipermutable in G if H has a supplement T in G such that H is X-permutable with every Sylow subgroup of T. Using the X-s-semipermutability of subgroups, the following results are obtained：①Let f be a saturated formation containing all supersoluble groups, G a group and X a soluble normal subgroup of G. Then GE f if and only if there exits H G such that G/H∈f and every maximal subgroup of every Sylow subgroup of H is X-s-semipermutable in G.②Let f be a saturated formation containing all supersoluble groups. Suppose that X is a soluble normal subgroup of G and H G, if G∈f and every maximal subgroup of every Sylow subgroup of F（H） is X-s-semipermutable in G, then G∈f.③Let G be a group, p the smallest prime divisor of ｜G｜ and X a p-soluble normal subgroup of G. Suppose that H G, if G/H is p-nilpotent, G is A4-free and every 2-maximal subgroup of any Sylow p-subgroup of H is X-s-semipermutable in G, then G is p-nilpotent.