For a complex representation $V$ of a finite group $G$, define the action of $G$ on the dual representation $V^{*}$. If $\alpha$ denotes the character of $V$, compute the character $\beta$ of $V^{*}$.

[Your formula should express $\beta(g)$ just in terms of the character $\alpha$.]

Using your formula, how can you tell from the character whether a given representation is self-dual, that is, isomorphic to the dual representation?

Let $V$ be an irreducible representation of $G$. Show that the trivial representation occurs as a summand of $V \otimes V$ with multiplicity either 0 or 1 . Show that it occurs once if and only if $V$ is self-dual.

For a self-dual irreducible representation $V$, show that $V$ either has a nondegenerate $G$-invariant symmetric bilinear form or a nondegenerate $G$-invariant alternating bilinear form, but not both.

If $V$ is an irreducible self-dual representation of odd dimension $n$, show that the corresponding homomorphism $G \rightarrow G L(n, \mathbf{C})$ is conjugate to a homomorphism into the orthogonal group $O(n, \mathbf{C})$. Here $O(n, \mathbf{C})$ means the subgroup of $G L(n, \mathbf{C})$ that preserves a nondegenerate symmetric bilinear form on $\mathbf{C}^{n}$.