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What is the difference between a $G$-module homomorphism and a $G$-linear map? My understanding is that both are linear maps between vector spaces (say, from $V$ to $W$) and both preserve the action of the group $G$ (i.e., for all $g$ in $G$, $\rho(g*v)=g*\rho(v)$ for all $v$ in $V$).

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No difference, just different words.

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