# On X-ss-permutable Subgroups of Finite Groups

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Let â„± be a class of groups and let G be a finite group. We call a set Î£ of subgroups of G a covering subgroup system of G for â„± (or directly an â„±-covering subgroup system of G) if G âˆˆ â„± whenever every subgroup in Î£ is in â„±. We give some covering subgroup...

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We study the structure of a finite group G under the assumption that every subgroup of prime order p and every cyclic subgroup of order 4 are contained in the hypercenter or the generalized hypercenter of the normalizer NG (P) of Sylow p-subgroup P of G.

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It is a well-known fact that subgroups of a group can give information of whole group, especially normal subgroups in a group greatly influence on the structure of the group. For example, a nilpotent group is a direct product of Sylow subgroups; a solvable group is a group with a normal series...

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- The Influence of X-s-semipermutable Subgroups on the Structure of Finite Groups. Hao, L. P.; Zhang, X. J.; Yu, Q. // Southeast Asian Bulletin of Mathematics;2009, Vol. 33 Issue 3, p421
Let X be a non-empty subset of a group G. A subgroup A of G is said to be X-s-semipermutable in G if A has a supplement T in G such that for every Sylow subgroup Tp of T, there exists an element x âˆˆ X such that ATpx = Tpx A. In this paper, we use X-s-semipermutable subgroup to determine the...

- ON WEAKLY SS-QUASINORMAL AND HYPERCYCLICALLY EMBEDDED PROPERTIES OF FINITE GROUPS. TAO ZHAO // International Journal of Group Theory;2014, Vol. 3 Issue 4, p17
A subgroup H is said to be s-permutable in a group G, if HP = PH holds for every Sylow subgroup P of G. If there exists a subgroup B of G such that HB = G and H permutes with every Sylow subgroup of B, then H is said to be SS-quasinormal in G. In this paper, we say that H is a weakly...

- Second Maximal Subgroups of a Sylow p-Subgroup and the p-Nilpotency of Finite Groups. Xu, Y.; Li, X. // Ukrainian Mathematical Journal;Oct2014, Vol. 66 Issue 5, p775
A subgroup H of a group G is said to be weakly s-semipermutable in G if G has a subnormal subgroup T such that HT = G and H âˆ© T â‰¤ $$ {H}_{\overline{s}G} $$, where $$ {H}_{\overline{s}G} $$ is the subgroup of H generated by all subgroups of H that are s-semipermutable in G. The main...

- Finite Groups in Which Every Non-abelian Subgroup Is s-permutable. Luo, X. L.; Guo, X. Y. // Southeast Asian Bulletin of Mathematics;2009, Vol. 33 Issue 6, p1143
A subgroup H of a finite group G is said to be s-permutable in G if H permutes with every Sylow subgroup of G. In this paper we investigate the structure of non-nilpotent groups in which every non-abelian subgroup is s-permutable.