Exercise 2.1.Let \(G\) be a finite group, \(\rho\) be an irreducible representation of \(G\), let \(\chi\) be its character, and let \(g\in G\). Show that:

1. (i) \(|\chi(g)|\leq |\chi(1)|\);
2. (ii) \(\chi(g)=\chi(1)\) if and only if \(g\in\ker\rho\);
3. (iii) \(|\chi(g)|=|\chi(1)|\) if and only if \(g\) lies in the centre of \(G/\ker\rho\).

Exercise 2.2. Let \(M\) be a \(kG\)-module with character \(\chi\). Prove that the characters \(\chi_{\Lambda^2(M)}\) and \(\chi_{S^2(M)}\) of \(\Lambda^2(M)\) and \(S^2(M)\) are\[ \chi_{\Lambda^2(M)}(x)=\frac12(\chi(x)^2-\chi(x^2))\]and\[ \chi_{S^2(M)}(x)=\frac12(\chi(x)^2+\chi(x^2))\]respectively. (Hint: consider a basis for the matrix representation of the group element consisting of eigenvectors.)

Exercise 2.3. Let \(V\) be a vector space over a field \(k\), and let \(G\) be a subgroup of \(\mathrm{GL}(V)\). Let \(f(-,-)\) be a \(G\)-invariant bilinear form. Show that the left and right radicals of \(f(-,-)\) are \(G\)-invariant.

Exercise 2.4. Let \(V\) be a \(\mathbb{C} G\)-module with associated character \(\chi\). Show that the character associated to the module \(({}_\mathbb{R} V)_\mathbb{C}\) is \(\chi+\chi^*\).

Exercise 2.5. Let \(G\) be a finite group. Show that\[ \sum_{\chi\in\mathrm{Irr}(G)} \chi(1)\cdot \nu(\chi)\]is equal to the number of elements of \(G\) that square to the identity. This is sometimes called the Frobenius-Schur count of involutions.

Exercise 2.6. Let \(G\) be a finite group. Analogously to \(\nu(\chi)\), let \(\nu_m(\chi)\) be defined by\[ \nu_m(\chi)=\frac{1}{|G|}\sum_{g\in G} \chi(g^m).\]Prove that\[ \sum_{\chi\in\mathrm{Irr}(G)} \chi(1)\cdot \nu_m(\chi)\]is equal to the number of elements of \(G\) whose \(m\)th power is the identity. (Depending on how you proved Exercise 2.5, your proof might generalize immediately.)

Exercise 2.7. Let \(G\) and \(H\) be finite groups and let \(k\) be an algebraically closed field. Prove that, as \(k(G\times H)\)-modules, we have \(k(G\times H)\cong kG\otimes kH\). Using the Jordan-Hölder theorem, deduce that every composition factor of \(k(G\times H)\) is of the form \(V\otimes W\), where \(V\) is the inflation of a simple \(kG\)-module and \(W\) is the inflation of a simple \(kH\)-module.

Deduce that every simple \(k(G\times H)\)-module is of the form \(V\otimes W\), using Exercise 1.11.