# Question for the mathematical audience

Published: Thu 20 April 2006

I have now been staring at this particular sentence for way too long, and thus will start using any and all communication lines I can find to get assistance. Either I'm being way too stupid, or the author neglects to mention some salient detail.

Setup: $$\phi\colon G^\prime\to G$$ is a group homomorphism, $$A\in kG-\operator{mod}$$, $$A^\prime\in kG^\prime-\operator{mod}$$. $$A$$ can be given the structure of a $$kG^\prime$$-module by pulling back through $$\phi$$, i.e. we define $$g^\prime a:=\phi(g^\prime)a$$ for $$g^\prime\in G^\prime$$ and $$a\in A$$.

So far it's all crystal clear for me. However, it then turns out that we're highly interested in using a morphism $$f\in\operator{Hom}_G(A,A^\prime)$$ and I cannot for the life of me find out how such beasts are guaranteed to exist. If it where $$f\in\operator{Hom}_{G^\prime}(A,A^\prime)$$, I wouldn't have any problems with it; but then the stuff I need/want to do with it don't work out.