Character Theory

Suppose \(\sigma \) and \(\tau \) are two elements of \(A_5\) such that, for every finite dimensional complex representation \(\rho \) of \(S_5\), \(\chi _\rho (\sigma )=\chi _\rho (\tau )\) (here \(\chi _\rho \) denotes the character of \(\rho \)). Are \(\sigma \) and \(\tau \) conjugate in \(A_5\)?

Let \(G\) be the group of symmetries of an equilateral triangle. Show that the 2 dimensional representation (as symmetries of the triangle) is irreducible by calculating the character table.

Let \(D_8\) be the group of symmetries of a square. Show that the 2 dimensional representation (as symmetries of the square) is irreducible by calculating the character table.