# When are two Representations equivalent?

Let $K$ be a field, $G$ group, $V$ and $W$ be finite dimensional $K[G]$ modules and $X$ and $Y$ be the representations afforded by $V$ and $W$ respectively.I need to show that $X$ and $Y$ are equivalent if and only when $V$ and $W$ are as $K[G]$ modules isomorphic . Any hints !

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What is your definition of equivalent? Giving the same homomorphism of $G$ into a matrix group up to conjugation? –  Brett Frankel Apr 17 '12 at 21:06
yes thats what i mean. X=ZYZ^-1 for some Z . –  Theorem Apr 17 '12 at 21:09
Also, is this homework? If so, that's fine, just use the homework tag. –  Brett Frankel Apr 17 '12 at 21:09

Let $\varphi$ be an isomorphism from $V$ to $W$. Look at the action of $g$ on $W$ by writing it in terms of $\varphi$ and action on $V$. (Remember, the action of a ring on a module commutes with module homomorphisms). That should hopefully be enough of a hint. Let me know if you're still confused after trying to work this out.
EDIT: $V=\varphi(\varphi^{-1} (W))$. So if $v \in V$, $g\cdot v=g\cdot \varphi(\varphi^{-1} (v))=\varphi(g\cdot\varphi^{-1} (v))$ In other words, acting by $g$ is the same as mapping $V$ to $W$, acting by $g$ in $W$, and then mapping back to $V$. Writing $\varphi$ as the linear map $Z$, we get that for any $g\in G$, $X(g)=Z\cdot Y(g)\cdot Z^{-1}$