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 ${\pi }_e(G)$ be the set of the orders of elements in $G$. Then there exists a non-negative integer $k$ such that $k(G)=|{\pi }_e(G)|+k$. We call such groups to be $co(k)$ groups. This paper 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 $Z_3⋊Z_4$.

15 pages