Finite Groups with Conjugacy Classes Number One Greater than Its Same Order Classes Number

ABSTRACT

Let k(G) be the number of conjugacy classes of finite groups G and π _e (G) be the set of the orders of elements in G. Then there exists a non-negative integer k such that k(G) = |π _e (G)| + k. We call such groups to be co(k) groups. This article classifies all finite co(1) groups. They are isomorphic to one of the following groups: A ₅, L ₂(7), S ₅, Z ₃, Z ₄, S ₄, A ₄, D ₁₀, Hol(Z ₅), or Z ₃ ⋊ Z ₄.

Key Words:

• Conjugacy class
• Finite group
• Same order class

2000 MR Subject Classification:

• 20D60
• 20D05
• 20D10

ACKNOWLEDGMENTS

The project is supported by the National Natural Science Foundation (Grant No. 10171074), and the Natural Science Foundation of Chongqing Education Commission (KG051107).

Notes

The first row denotes the Out(N), where N is a non-Abelian simple group listed in the table. nA(m) denotes the conjugacy class of elements of order n. m denotes the order of centralizer of a representative of the conjugacy class.

Communicated by A. Olshanskiy.