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Mathematics 2019, 7(2), 158; https://doi.org/10.3390/math7020158

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

1
School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China
2
School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China
*
Author to whom correspondence should be addressed.
Received: 14 January 2019 / Revised: 3 February 2019 / Accepted: 3 February 2019 / Published: 10 February 2019
PDF [280 KB, uploaded 10 February 2019]

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 H-embedded subgroups; p-supersolubility; p-nilpotence finite groups; weakly H-embedded subgroups; p-supersolubility; p-nilpotence
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Zhang, L.; Huo, L.-J.; Liu, J.-B. Some New Applications of Weakly H-Embedded Subgroups of Finite Groups. Mathematics 2019, 7, 158.

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