I'm having a little trouble understanding (intuitively or otherwise) what is meant by a module over a group algebra FG (F a field, G a finite group) and how it relates to transformations.

For example, how do I go about thinking about this question:

Identify

with the subgroup of

which fixes 4 (

is of course the symmetric group on

letters). Then

acts on

by conjugation (???). Find the character of the linearization

as a

-module, and determine the multiplicities of each simple

-module in

.

I would be very very grateful for any light anyone could shed on this question; especially any general comments about FG-modules and representations.

Thanks in advance!