Abstract
A subgroup H of a finite group G is said to be c*-supplemented in G if there exists a subgroup K such that G = HK and H ⋂ K is permutable in G. It is proved that a finite group G that is S 4-free is p-nilpotent if N G (P) is p-nilpotent and, for all x ∈ G\N G (P), every minimal subgroup of \(P \cap P^x \cap G^{\mathcal{N}_p } \) is c*-supplemented in P and (if p = 2) one of the following conditions is satisfied: (a) every cyclic subgroup of \(P \cap P^x \cap G^{\mathcal{N}_p } \) of order 4 is c*-supplemented in P, (b) \( [\Omega _2 (P \cap P^x \cap G^{\mathcal{N}_p } ),P] \leqslant Z(P \cap G^{\mathcal{N}_p } )\), (c) P is quaternion-free, where P a Sylow p-subgroup of G and \(G^{\mathcal{N}_p } \) is the p-nilpotent residual of G. This extends and improves some known results.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1011–1019, August, 2007.
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Wei, H., Wang, Y. c*-Supplemented subgroups and p-nilpotency of finite groups. Ukr Math J 59, 1121–1129 (2007). https://doi.org/10.1007/s11253-007-0073-5
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DOI: https://doi.org/10.1007/s11253-007-0073-5