On PC-subgroups Of Finite Groups

A subgroup H of G is called a PC-subgroup if H is permutable with at least one conjugate of any subgroup of G,that is,for every subgroup K of G,there exists an element x∈G such that HK^x = K^xH.A subgroup H of G is a PCM-subgroup(or H is PCM in G) if H is permutable with at least one conjugate of each maximal subgroup of G,that is,for any maximal subgroup M of G,there exists an element x∈G such that HM^x = M^xH.A subgroup H of G is a PCSM-subgroup of G(or H is PCSM in G) if H is permutable with at least one conjugate of each maximal subgroup of any Sylow subgroup of G,that is,for each maximal subgroup M of any Sylow subgroup of G,there exists an element x∈G such that HM^x = M^xH.The aim of this paper is to investigate the influence of some certain PCM-subgroups and PCSM-subgroups on finite groups,and then to obtain the following main results.(1) Let G be a finite group and F be a saturated formation containing all supersolvable groups.Then G∈F if and only if one of the following statements is hold:(â…°) There exists a normal subgroup N of G such that G/N∈F and every cyclic subgroup of N of prime power order is PCM in G.(â…±) There exists a solvable normal subgroup N of G such that G/N∈F and every maximal subgroup of any Sylow subgroup of N is PCM in G.(â…²) There exists a solvable normal subgroup N of G such that G/N∈F and every cyclic subgroup of any Sylow subgroup of F(N) is PCM in G.(â…³) There exists a solvable normal subgroup N of G such that G/N∈F and every maximal subgroup of any Sylow subgroup of F(N) is PCM in G. (â…´) There exists a solvable normal subgroup N of G such that G/N∈F and every maximal subgroup of F(N) is PCM in G. Basically,we may get the following results(2)-(6) as corollaries.(2) The group G is supersolvable if and only if all cyclic subgroups of prime power order of G are PCM in G.(3) Let G be a solvable group.Then G is supersolvable if and only if all maximal subgroups of every Sylow subgroup of G are PCM in G.(4) Let G be a solvable group.Then G is supersolvable if and only if all cyclic subgroups of every Sylow subgroup of F(G) are PCM in G.(5) Let G be a solvable group.Then G is supersolvable if and only if all maximal subgroups of every Sylow subgroup of F(G) are PCM in G.(6) Let G be a solvable group.Then G is supersolvable if and only if all maximal subgroups of F(G) are PCM in G.(7) Let G be a finite group.Then all Sylow subgroups of G are PCM-subgroups if and only if G/Sol(G) =1 or L₂(7).(8) Let G be a solvable group.Then G is supersolvable if all Sylow subgroups of G are PCSM in G.(9) Let G be a solvable group.If all 2-maximal subgroups of G are PCSM in G, Then G/Fit(G) is abelian.