Having trouble working with modules

wizard147

Hi guys,

Basically I'm playing around with modules at the moment, and I can't work out why we cant have the group of integers as an F-module (F a field), where the left action is the identity.

i.e F x Z ----> Z

where we have f.z = z

f in F, z in Z

If this were possible, then Z would be a vector space wouldn't it, this is probably a stupid question but would be grateful if somebody could point out where I'm going wrong, i've been trying to work it out for hours.

Thanks!

C

micromass

Does the axiom

[tex](f+f^\prime)z=fz+f^\prime z[/tex]

Still hold??

What about 0z=0 ??

wizard147

Ahhh, so simple!

Thank you so much, I think i'll noodle around with them a bit more so I can understand them better.

Really appreciate it.

C

wizard147

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

micromass

Yes, that seems right indeed!!

wizard147

Awesome, cheers Micromass!

C

wizard147

Ah wait no,

What I said that (g+h) isnt a basis element is wrong. It is an element of G and thus a basis element. Therefore we have

(g+h).v = g.v +h.v

implies v=v+v

So I'm still stuck as to how we can get left action to be the identity.

thought we almost had it!

C