Terminology issue regarding modules and representations

[Office_Shredder]

Homework Statement

Given a field F, FS₄ is a group algebra... we have a representation X that maps FS₄ to 3x3 matrices over (presumably) F. Let V denote the FS₄ module corresponding to X... do stuff. My question is, what the heck is V supposed to be?

I assumed that V is F³, but that seemingly contradicts what the question wants me to do. So is V the free module of the representation of FS₄ over itself (and then why bother having a representation)? I can't think of what else it could be

The full question can be found at

http://www.maths.ox.ac.uk/courses/2008/part-b/b2-algebra/b2a-introduction-representation-theory/materialsheet 5, question 3

EDIT: I was wrong... if V=F³ the question's conclusion is correct. For some reason I was associating the 4 in S₄ with F instead, and thought it had characteristic 4... I noticed my error when I realized a) it has characteristic 0 b) 4 isn't the characteristic of a field. Good job by me wasting an hour working in a non-existent field

 

[morphism]

Well the representation gives us a homomorphism from S_4 into GL(F^3). The induced action of the algebra FS_4 on F^3 induces an FS_4-module structure on F^3. So yes, V=F^3.