1. Assume that the field is algebraically closed and has zero characteristic, G is finite and representations are finite-dimensional.
Show that this statement is true under the above assumptions:
"Let p be an irreducible representation of G, and q be an irreducible representation of H. Is it always true that the exterior tensor product of p and q is an irreducible representation of G X H?"
2. Let p: G -> GL(V) be a representation. Show that each irreducible subrepresentation of V has multiplicity 1 iff EndG(V) is a commutative ring.
There are two proofs here regarding irreducible representations.