Is this really a module?

[Office_Shredder]

Homework Statement

K is a field with finite characteristic p, G is a finite group, and W is a set that G acts on transitively (so for all x,y in W, there exists g s.t. gx=y). It then says consider M=KW the permutation module.

What is KW supposed to mean? I know for a group G that KG is the group algebra, but we don't know that W is a group (in fact, it probably isn't). Furthermore, what ring is intended to be used for multiplication? I'm confused out of my mind. I've looked back in my lecture notes so far but haven't seen anything to resolve the issue

The Attempt at a Solution

Asking here

 

[morphism]

I think in this setting the permutation module is the KG-module you obtain by letting G act on KW = set of formal linear combinations of elements of W with coefficients in K (which is basically the free K-module generated by W).

 

[Office_Shredder]

But G isn't a ring. Unless we just use formal addition in G to make it one?

 

[morphism]

G isn't; KG is.

This sort of stuff comes up when you talk about things like "G-modules". See http://planetmath.org/encyclopedia/GModule.html .

 

[Office_Shredder]

Oh, I misunderstood what you wrote originally. That makes sense now