Some New Applications of Weakly ?-Embedded Subgroups of Finite Groups

Li Zhang, Li-Jun Huo and Jia-Bao Liu
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Li Zhang: School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China
Li-Jun Huo: School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China
Jia-Bao Liu: School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China

Mathematics, 2019, vol. 7, issue 2, 1-10

Abstract: A subgroup H of a finite group G is said to be weakly H -embedded in G if there exists a normal subgroup T of G such that H G = H T and H ∩ T ∈ H ( G ) , where H G is the normal closure of H in G , and H ( G ) is the set of all H -subgroups of G . In the recent research, Asaad, Ramadan and Heliel gave new characterization of p -nilpotent: Let p be the smallest prime dividing | G | , and P a non-cyclic Sylow p-subgroup of G. Then G is p-nilpotent if and only if there exists a p-power d with 1 < d < | P | such that all subgroups of P of order d and p d are weakly H -embedded in G . As new applications of weakly H -embedded subgroups, in this paper, (1) we generalize this result for general prime p and get a new criterion for p -supersolubility; (2) adding the condition “ N G ( P ) is p -nilpotent”, here N G ( P ) = { g ∈ G | P g = P } is the normalizer of P in G , we obtain p -nilpotence for general prime p . Moreover, our tool is the weakly H -embedded subgroup. However, instead of the normality of H G = H T , we just need H T is S -quasinormal in G , which means that H T permutes with every Sylow subgroup of G .

Keywords: finite groups; weakly ?-embedded subgroups; p -supersolubility; p -nilpotence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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