PROBLEM SET # 3MATH 252Due March 1.For the first three problems we assume that q is a power of an odd prime number,Fqis a finite field with q elements and GL(2, Fq) is the group of all invertible matriceswith coefficients in Fq. Below I collect the information about irreducible representa-tions of GL(2, Fq) which we o bta ined in class. It is useful for this assignment.Recall that the conjugacy classes of GL(2, Fq) are of four types (for each class wegive a representat ive):Cx: (x00x), x ∈ F∗qDx: (x01x), x ∈ F∗qHx: (x00y), x, y ∈ F∗qKζ: (xyεyx), ζ ∈ Fq2\ FqIrreducible representations are1) 1-dimensional: detα, (here α : F∗q→ C∗is a character).2) q-dimensional; Vα(here α : F∗q→ C∗is a character).χVα=α(x2)q, 0, α(xy), −α(ζq+1);3) (q + 1)-dimensional Wα,β, (here α, β : F∗q→ C∗are characters, α 6= β)χWα,β= (α(x)β(x)(q + 1), α(x)β(x), α(x)β(y) + α(y)β(x), 0) .We have Wα,β≃ Wβ,α.4) (q − 1)-dimensional Xφ(here φ : F∗q2→ C∗is a character such that φ 6= φq)),with the characterχXφ= (φ(x)(q − 1), −φ(x), 0, −(φ(ζ) + φ(ζq))) .We have Xφ≃ Xφq.1. Let P GL(2, Fq) be the quotient by the center. Calculate the order ofP GL(2, Fq) and show that P GL(2, F3) is isomorphic to the symmetric group S4.(Hint: consider the action of P GL(2, F3) o n P1(F3).)2. Calculate the character table for P GL(2, Fq). (Hint: use the character tablefor GL(2, Fq).)Date: February 17, 2011.12 PROBLEM SET # 3 MATH 2523. Let q > 3. Use the previous problem to show that the only non-trivial propernormal subgroup of P GL(2, Fq) is P SL(2, Fq) which is the quotient of SL(2, Fq) bythe center. Use this to prove that P SL(2, Fq) is simple.4. Let G be a finite gr oup, r be the number of conjugacy classes in G and s be thenumber of conjugacy classes in G preserved by the involution g → g−1. Prove thatthe number of irreducible representations of G over R is equal