A note on the conjugacy classes of non-cyclic subgroups

ABSTRACT

Let G be a finite group and δ(G) denote the number of conjugacy classes of all non-cyclic subgroups of G. The symbol π(G) denotes the set of the prime divisors of |G|. In [7], Meng and Li showed the inequality δ(G)≥2^|π(G)|−2, where G is non-cyclic solvable group. In this paper, we describe the finite groups G such that δ(G) = 2^|π(G)|−2. Another aim of this paper would show δ(G)≥M(G)+2 for unsolvable groups G and the equality holds ⇔G≅A₅ or SL(2,5), where M(G) denotes the number of conjugacy classes of all maximal subgroups of G.

KEYWORDS:

• Conjugacy class
• cyclic subgroups
• maximal groups

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D10
• 20D20

Acknowledgments

The authors are grateful to the referee, who provided their valuable suggestions and detailed report.