A constructive approach: From local subgroups to new classes of finite groups

Year 2023, , 420 - 425, 31.03.2023

Zhencai Shen , Baoyu Zhang , Haonan Jıang

https://doi.org/10.15672/hujms.1114323

Abstract

Let G be a finite group and S be a proper subgroup of G. A group G is called an S-(S-quasinormal)-group if every local subgroup of G is either an S-quasinormal subgroup or conjugate to a subgroup of S. The main purpose of this construction is to demonstrate a new way of analyzing the structure of a finite group by the properties and the number of conjugacy classes of its local subgroups.

Keywords

local subgroups, simple groups, solvable groups, S-quasinormal subgroups

Supporting Institution

Natural Science Foundation of China

Project Number

No. 12071181

References

• [1] R.K. Agrawal, Finite groups whose subnormal subgroups permute with all Sylow subgroups, Proc. Amer. Math. Soc. 47 (1), 77–83, 1975.
• [2] A. Beltrán, On powers of conjugacy classes in a finite group, J. Group Theory 25 (5), 965–971, 2022.
• [3] A. Beltrán and M.J. Felipe, Cosets of normal subgroups and powers of conjugacy classes, Math. Nachr. 294, 1652–1656, 2021.
• [4] J.D. Dixon and B. Mortimer, Permutation Groups, Springer-Verlag, New York, 1996.
• [5] B. Huppert, Endliche Gruppen, Springer-Verlag, Berlin, Heidelberg, New York, 1967.
• [6] Q. Jiang and C. Shao, Primary and biprimary class sizes implying nilpotency of finite groups, Turkish J. Math. 40 (2), 389–396, 2016.
• [7] O.H. Kegel, Sylow-Gruppen und Subnormalteiler endlicher Gruppen, Math. Z. 78, 205–221, 1962.
• [8] S. Li, Z. Shen, J. Liu and X. Liu, The influence of SS-quasinormality of some subgroups on the structure of finite groups, J. Algebra 319, 4275–4287, 2008.
• [9] D.J.S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York, 1996.
• [10] C. Shao and Q. Jiang, Finite groups with three conjugacy class sizes of primary and biprimary elements, Turkish J. Math. 39 (3), 346–355, 2015.
• [11] S. Srinivasan, Two sufficient conditions for supersolvability of finite groups, Israel J. Math. 35, 210–214, 1980.
• [12] J.G. Thompson, Normal p-complements for finite groups, Math. Z. 72, 332–354, 1959/1960.
• [13] J.G. Thompson, Normal p-complements for finite groups, J. Algebra 1, 43–46, 1964.
• [14] J.G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74, 383–437, 1968.