Let G be a finite group.A subgroup H is called SS-quasinormal in G if there is a subgroup B of G such that G=HB,and HP=PH holds for all prime p∈π(B)and P∈Sylp(B).The structures of finite groups with SS-quasinormality of primary subgroups are studied.Some new criteria of p-nilpotent group are given by using induction on the order of G and counterexample of minimal order.
SS-quasinormal subgroupp-nilpotent groupp-supersolvable groupinductioncounterexample of minimal order
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