Exercise 4.1. Let \(H\) be a subgroup of \(G\) of index \(n\), and let \(t_1,\dots,t_n\) denote a right transversal to \(H\) in \(G\). If \(g\in G\), then there exists a function \(\sigma:\{1,\dots,n\}\to\{1,\dots,n\}\) and elements \(h_i\in H\) for \(1\leq i\leq n\) such that\[ t_ig=h_it_{i\sigma},\]since the \(t_i\) form a right transversal.

Prove that \(\sigma\) is a permutation of \(\{1,\dots,n\}\). Furthermore, show that if \(g\) and \(g'\) correspond to permutations \(\sigma\) and \(\sigma'\), then \(gg'\) corresponds to \(\sigma\sigma'\).

Exercise 4.2. Let \(G\) be a finite group and let \(K\) and \(H\) be subgroups of \(G\) with \(K\leq H\leq G\). If \(M\) is a \(kK\)-module, show that\[ M{\uparrow^G}\cong (M{\uparrow^H}){\uparrow^G}.\]

Exercise 4.3. Let \(G\) be a finite group, \(H\) be a subgroup of \(G\), let \(M\) be a \(kG\)-module and \(N\) a \(kH\)-module. Show that\[ (N\otimes M{\downarrow_H}){\uparrow^G}\cong N{\uparrow^G}\otimes M.\]

Exercise 4.4. Let \(G\) be a finite group, \(H\) be a subgroup of \(G\), and let \(M\) be a \(kH\)-module and \(N\) a \(kK\)-module. Show that\[ M{\uparrow^G}\otimes N{\uparrow^G}\cong \bigoplus_{t\in T} (M^t{\downarrow_{{H^t\cap K}}}\otimes N{\downarrow_{{H^t\cap K}}}){\uparrow^G},\]where \(T\) is a set of \((H,K)\)-double coset representatives.

Exercise 4.5. Let \(G\) be a non-abelian \(p\)-group of order \(p^3\).

1. (i) Determine the character group \(X(G)\).
2. (ii) Prove that there are \(p^2+p-1\) conjugacy classes and irreducible characters of \(G\).
3. (iii) Using the orthogonality relations and tensor products with \(1\)-dimensional modules, determine the character table of \(G\).