On Solvability of Finite Groups with Few Non-Normal Subgroups

Abstract

For a finite group G, let v(G) denote the number of conjugacy classes of non-normal subgroups of G and vc(G) denote the number of conjugacy classes of non-normal noncyclic subgroups of G. In this paper, we show that every finite group G satisfying v(G) ≤2|π(G)| or vc(G) ≤ |π(G)| is solvable, and for a finite nonsolvable group G, v(G) = 2|π(G)| +1 if and only if G ≅ A ₅.

Key Words:

• Noncyclic subgroups
• Non-normal subgroups
• Solvable groups

2010 Mathematics Subject Classification:

• 20D10
• 20D25

ACKNOWLEDGMENTS

The authors are grateful to the referee who gives valuable comments and suggestions.

Notes

Communicated by V. A. Artamonov.