On non-normal non-abelian subgroups of finite groups

Document Type : Research Paper


School of Mathematics and Information Science‎, ‎Yantai University‎, ‎Yantai 264005‎, ‎China.


‎In this paper we prove that a finite group $G$ having at most three‎ ‎conjugacy classes of non-normal non-abelian proper subgroups is‎ ‎always solvable except for $G\cong{\rm{A_5}}$‎, ‎which extends Theorem 3.3‎ ‎in [Some sufficient conditions on the number of‎ ‎non-abelian subgroups of a finite group to be solvable‎, ‎Acta Math‎. ‎Sinica (English Series) 27 (2011) 891--896.]‎. ‎Moreover‎, ‎we show that a‎ ‎finite group $G$ with at most three same order classes of non-normal‎ ‎non-abelian proper subgroups is always solvable except for $G\cong‎{A_5}$‎.


Main Subjects

V.A. Belonogov, Finite groups with three classes of maximal subgroups, Mat. sb. 131 (1986), no. 2, 225--239.
J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
S. Li, J. Shi and X. He, Some necessary and sufficient conditions of abelian groups and cyclic groups (Chinese), Guangxi Sciences 13 (2006), no. 1, 1--3.
G.A. Miller and H.C. Moreno, Non-abelian groups in which every subgroup is abelian, Trans. Amer. Math. Soc. 4 (1903), no. 4, 398--404.
J. Shi and C. Zhang, The inuence of some quantitative properties of abelian subgroups on solvability of finite groups, Comm. Algebra 39 (2011), no. 10, 3916--3922.
J. Shi and C. Zhang, Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable, Acta Math. Sin. (Eng. Ser.) 27 (2011), no. 5, 891--896.
J. Shi and C. Zhang, Finite groups in which the number of classes of non-nilpotent proper subgroups of the same order is given (Chinese), Chinese Ann. Math. Ser. A 32 (2011), no. 6, 687--692.
H. Zassenhaus, A group-theoretic proof of a theorem of Maclagan-Wedderburn, Proc. Glasgow Math. Assoc. 1 (1952) 54--63.