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Having trouble working with modules

 Quote by micromass Does the axiom $$(f+f^\prime)z=fz+f^\prime z$$ Still hold?? What about 0z=0 ??
Another question for you :),

I've been playing around with these and had a look at a bit of representation theory.

I was looking at group algebra's, where G is a finite group, and \mathbb{C} is our field.

I was trying to find a C[G] module, V, that has a left action is the basis of C[G] (i.e. group elements of G) on V as the identity map,

i.e. g.v=v where g \in{G}

I wasn't sure if the axiom (g+h).v = g.v + h.v held, but I think its to do with the fact that g+h isn't a basis element and thus

(g+h).v \neq v

and that the axiom holds trivially as our left action is a group homomorphism

i.e. (g+h).v = g.v + h.v by definition

am I right in saying this, otherwise I can't work out how we get the identity rep for group algebra's under the correspondence theorem in representation theory.

Thanks in advance

Sorry if this is unclear, just say if you can't work out what i'm trying to say

C