Originally Posted by

**AtticusRyan** 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 $\displaystyle S_3$ with the subgroup of $\displaystyle S_4$ which fixes 4 ($\displaystyle S_n$ is of course the symmetric group on $\displaystyle n$ letters). Then $\displaystyle S_3$ acts on $\displaystyle S_4$ by conjugation (???). Find the character of the linearization $\displaystyle \mathbb{C} S_4$ as a $\displaystyle \mathbb{C} S_3$-module, and determine the multiplicities of each simple $\displaystyle \mathbb{C} S_3$-module in $\displaystyle \mathbb{C} S_4$.

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!