On conjugacy classes of SL$(2,q)$

Abstract

Let SL(2,q) be the group of 2X2 matrices with determinant one over a finite field F of size q. We prove that if q is even, then the product of any two noncentral conjugacy classes of SL(2,q) is the union of at least q-1 distinct conjugacy classes of SL(2,q). On the other hand, if q>3 is odd, then the product of any two noncentral conjugacy classes of SL(2,q) is the union of at least (q+3)/2 distinct conjugacy classes of SL(2,q).