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I need the right answers for Q3 (a and b) with explanation.
1.13 Exercises 1. An inversion in π = x1,x2,..., n E Sn (one-line notation) is a pair Ti, Tj such that i<j and ₁>₁. Let inv 7 be the number of inversions of T. (a) Show that if can be written as a product of k transpositions, then k = invπ (mod 2). (b) Use part (a) to show that the sign of T is well-defined. If group G acts on a set S and s E S, then the stabilizer of s is G, = {ge G gs=s}. The orbit of s is O. = {gs ge G}. (a) Prove that G, is a subgroup of G. (b) Find a bijection between cosets of G/G, and elements of Os (c) Show that |O| = |G|/IG, and use this to derive formula (1.1) for |Kgl. Let G act on S with corresponding permutation representation CS. Prove the following. (a) The matrices for the action of G in the standard basis are permu- tation matrices. (b) If the character of this representation is x and g € G, then x(g) = the number of fixed points of g acting on S. 4. Let G be an abelian group. Find all inequivalent irreducible represen- tations of G. Hint: Use the fundamental theorem of abelian groups. 5. If X is a matrix representation of a group G, then its kernel is the set N = {g € G: X(g) = 1}. A representation is faithful if it is one-to-one. (a) Show that N is a normal subgroup of G and find a condition on N equivalent to the representation being faithful. (b) Suppose X has character x and degree d. Prove that g E N if and only if x(g) = d. Hint: Show that x(9) is a sum of roots of unity. Show that for the coset representation, N = nigiHg¹, where the 9₁ are the transversal. (d) For each of the following representations, under what conditions are they faithful: trivial, regular, coset, sign for Sn, defining for Sn, degree 1 for Cn? (e) Define a function Y on the group G/N by Y(gN) = X(g) for ON SCHN