Some New Applications of Weakly H-Embedded Subgroups of Finite Groups
School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China
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 -embedded in G if there exists a normal subgroup T of G such that and , where is the normal closure of H in G, and is the set of all -subgroups of G. In the recent research, Asaad, Ramadan and Heliel gave new characterization of p-nilpotent: Let p be the smallest prime dividing , 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 such that all subgroups of P of order d and are weakly -embedded in G. As new applications of weakly -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 “ is p-nilpotent”, here is the normalizer of P in G, we obtain p-nilpotence for general prime p. Moreover, our tool is the weakly -embedded subgroup. However, instead of the normality of , we just need is S-quasinormal in G, which means that permutes with every Sylow subgroup of G.
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).
Share & Cite This Article
MDPI and ACS Style
Zhang, L.; Huo, L.-J.; Liu, J.-B. Some New Applications of Weakly H-Embedded Subgroups of Finite Groups. Mathematics 2019, 7, 158.
Zhang L, Huo L-J, Liu J-B. Some New Applications of Weakly H-Embedded Subgroups of Finite Groups. Mathematics. 2019; 7(2):158.
Zhang, Li; Huo, Li-Jun; Liu, Jia-Bao. 2019. "Some New Applications of Weakly H-Embedded Subgroups of Finite Groups." Mathematics 7, no. 2: 158.
Show more citation formats
Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.
[Return to top]
For more information on the journal statistics, click here
Multiple requests from the same IP address are counted as one view.