dimension of a irreducible representation

Let G be a group and A an abelian subgroup of G. I want to prove that for each irreducible representation $p$ we have that $dim(p)leq [G:A]$.
I proved that for each subspace $p(A)-$invariant $W$, $W_0= sum_{g in G}p(g)(W)$ is a p(G)-invariant of V (hint of the professor) and that for a abelian finite group, each irreducible representation has dimension 1.
But how can i go on? I wish i can tell more of what i did in this question but i honestly have no idea how to proceed.I was thinking in the canonical decomposition of a representation, considering the subspaces W,but after some writing it seems non-sense.
Any help are welcome.
Ps:my representation theory course uses only knowlegde of theory of groups, not modules or other stuff.