Irreducible representations

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?" (my comment: the answer to this under the above assumption is of course yes and I have a way to show it but would like to see if I'm actually right. Thanks a lot!) 2. Let p: G -> GL(V) be a representation. Show that each irreducible subrepresentation of V has multiplicity 1 iff EndG(V) is a com... click for more

irreducible representations of a quaterion subgroup

Let G be the subgroup of quaternions of 8 elements, that contains ±1, ±i, ±j, ±k with relations i^2=j^2=k^2= −1, ij=k, jk=i, ki=j, ij=−ji, ik=−ki, jk=−kj. Classify irreducible representations of G over C.

Dodecahedron problem A_5 irreducibles

Consider the action of the group A_5 on the faces of a dodecahedron. Decompose the corresponding representation of A_5 into a sum of irreducibles and solve the problem by diagonilizing the interwining operator.

Matrix irreducible representations

Let G be the group of matrices 1 x y 0 1 z 0 0 1 where x, y, z are elements of the finite field F_5 . Classify irreducible representations of G over C.

Transitivity of induced representations

PLEASE SEE ATTACHMENT. A very standard proof in induced representation theory. can be found in any relevant book yet I have trouble unerstanding it. Please write the complete proof with details and everything. Thanks a lot. This will help me prepare for my exams.