Let V be a finite dimensional representation of a finite group G. Consider Hom(V,V) = Set of all Linear Transformations from V to V. Its clearly a vector space (under the natural operations).

Prove that : Hom(V,V) is an irreducible representation of G $\displaystyle \Leftrightarrow$ dim V = 1

The problem for me is that I cant understand how the group GL(Hom(V,V)). It looks extremely complicated.

Trying to prove: $\displaystyle \Leftarrow$

I assume ($\displaystyle \phi$,V) is the representation over the field F, dim V = 1. Then $\displaystyle Hom(V,V) \cong F$.

F is a one-dimensional vector space over itself and thus Hom(V,V) is one dimensional and hence irreducible.

To prove: $\displaystyle \Rightarrow$

I am struggling here. So any ideas?

Thank you,

Srikanth