On non-normal non-abelian subgroups of finite groups

author

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

Abstract:

‎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 $Gcong{rm{A_5}}$‎, ‎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 $Gcong‎{A_5}$‎.

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