In lecture we have seen the permutation representation of the symmetric group Sn, which gives a group homomorphism p: Sn → GL(C"). Recall that this is defined by p(g)v¡ = Ug(i), where U₁, ..., Un are the standard basis vectors of C". We'll be considering properties of this homomorphism. Exercise B (see below for markdown cell) Show that tr(p(g)) is equal to the number of x E {1,..., n} such that g(x) = x.

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Q: Chapter 2, exercise 6.4: Prove that in a group, the products ab and ba are conjugate elements.

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A: Subgroup

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Q: orthogonal group O(n) = {A E GL(n, R)| AA = A'A

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