The Influence Of Weakly Quasinormal Subgroups On The Structure Of Finite Groups

It is a well-known fact that properties of subgroups of a group influence the structure of the group. We define the concepts: weakly quasinormal subgroup and S -weakly quasinormal subgroup and come to some new criteria of solvability, supersolvability and nilpotence of finite groups under the assumptions that the groups have some kinds of weakly quasinormal or 5 -weakly quasinormal subgroups. Some results are the following :(1) LetG be a finite group having two maximal subgroups that are solvable and not conjugate inG. If the maximal subgroups are weakly quasinormal inG, then G is solvable.(2) LetG be a finite group. G is a solvable group if and only if G has a solvable maximal subgroup being weakly quasinormal in G and no section isomorphic to group A5 , PSL2 (7) and PSL3 (3).(3) Suppose| (G)| = k and G has k-1 distinct Sylow subgroups such that the subgroups and all their maximal subgroups are weakly quasinormal in G, then G is supersolvable.(4) Let G be a solvable group. If G satisfies one of the following conditions, then G is supersolvable: (i) Every subgroup of F(G)are weakly quasinormal in G; (ii) For p (F(G)), every cyclic subgroup of Sylow p -subgroups of G are weakly quasinormal inG.(5) LetG be aQCLT-group. IfG satisfies one of the following conditions, thenG is supersolvable: (i) All maximal subgroups of Sylow 2 -subgroups of G are weakly quasinormal inG; (ii) There exits a Sylow 2 -subgroup of G such that its derived subgroup is weakly quasinormal inG ; (iii) There exits a Hall 2' -subgroup of G being weakly quasinormal in G .Note: If we substitute the assumption that a finite group G satisfies the permutizer condition for the assumption in (5) that a finite group G is a QCLT-group, then the same result follows.(6) Assume thatG has a nilpotent maximal subgroup M such that M and its maximal subgroups are weakly quasinormal in G. If G has no section isomorphic to D, where D =, p and q are distinct primes, thenG is a nilpotent group.(7) Assume thatG has two nilpotent maximal subgroups not conjugate inG,which are weakly quasinormal inG, then G is nilpotent if and only if G has no section isomorphic to D, where D is identical with one in (6).