一类B_p群及其应用

ACLASSOFB_pGROUPSANDITSAPPLICATIONS

群Ｇ称为Ｂ＿ｐ群，如果Ｎ＿ｇ（Ｐ）为ｐ－幂零蕴含Ｇ为Ｐ－幂零。证明了以下结论：１）设Ｐ为有限群Ｇ的ｐ－Ｓｙｌｏｗ子群，如果Ｐ的每极小子群及４阶循环子群在Ｐ内拟正规，则Ｇ为Ｂ＿ｐ群２）设Ｐ为有限群Ｇ的Ｐ－Ｓｙｌｏｗ子群。如果Ｐ的极小子群及４阶循环子群在Ｎ＿Ｇ（Ｐ）内拟正规，且其每元与Ｎ＿Ｇ（Ｐ）的每ｑ－元（ｑ＜ｐ）可交换相乘，则Ｇ为Ｐ－幂零。３）有限群Ｇ若有正规子群Ｎ，使Ｇ／Ｎ∈，又对每Ｐ∈Ｓｙｌ（Ｎ）．均有Ｐ的极小于群及４阶循环子群在Ｎ＿Ｇ（Ｐ）内拟正规，则Ｇ∈。其中为包含超可解群系的饱和群系。

Call a p-subgroup P of a group H isv- regular in H. if every minimal subgroup and cyclic sub-group of order 4 of P are quasinormal in H,The following conclusion is proved:1)If the Sylowp-subgroup P of a finite group G is σ-regular in C,thenG is a B, group.2)Assume that P is a Sylow p-subgroup of a finite group G,If P is v-regular in N_G(p)and ev-ery element of P permuts with every q-element(q<P)of N_G (p ). then G isp- nilpotent.3) If a finite group G has a normal subgroup N such that G/N∈and each P∈SylN is σ-regu-lar in N_G(P).then G∈.where,is a saturated formtion containing the formation of supersolv-able grotups.