Complex representations of finite general linear groups

In this thesis, we determine complex irreducible representations of GL(2,K), the group of 2 by 2 invertible matrices over a finite fi eld K. Actually, this is done by Ilya Piatetski-Shapiro in 1983. In his article [1], Shapiro classi fies the irreducible representations of the group GL(2,K) by using the de finition of induced module depends as a space of functions. The aim of this thesis is to rewrite the article using the induction module defi nition constructed by a tensor product. We start the thesis by reminding some basic de finitions and theorems related to our topic. Then we determine the commutator subgroup of GL(2,K) and introduce some special subgroups of GL(2,K). The number of irreducible representations of a finite group is equal to the number of conjugacy classes of that group. Hence we calculate the conjugacy classes of GL(2,K). We determine irreducible representations of GL(2,K) through irreducible representations of the subgroups of it and quotient groups.