Where is the module in James and Liebeck's FG module

Question

On p. 39 of James and Liebeck's book "Representations and Characters of Groups" The term FG module is defined for F a field, G a group V a vectors space and an action of G on V is defined.

Where is the module in all this? My understanding is that a module is a vector space with multiplication by a ring (instead of a field). A group is not a Ring although it operates on V.

Where is a classically defined module in all this? Perhaps FG module is just a term of art

Answer 1

If $R$ is a ring, and $G$ is a group, then the group ring $\pmb{RG}$ is the ring of formal sums $$ RG \stackrel{\text{def}}{=} \bigg\{\widetilde\sum_{g\in G}r_gg : r_g\in R,\ g\in G\bigg\}, $$ where $\widetilde\sum$ means that all but finitely many summands are $0$. Addition is defined componentwise, and multiplication is handled like it is for polynomials.

Specializing to the case where $R = F$ is a field, $FG$ is still a ring, and if $V$ is an $F$-vector space on which $G$ acts, then $V$ is naturally an $FG$-module, where we define the module action for $\widetilde \sum_{g\in G}r_gg\in FG$ and $v\in V$ by $$ \bigg(\widetilde \sum_{g\in G}r_gg\bigg)\cdot v\ \stackrel{\text{def}}{=} \ \widetilde\sum_{g\in G}r_g(g\cdot v). $$ Note the $\cdot$ on the right-hand side of this equation is the action of $G$ on $V$.

Answer 2

The answer by Alex Ortiz is really great and was very helpful for me, but I think it’s possible to more directly address the original question: why does a $FG$-module (for example, as defined by James & Liebeck) qualify as a module?

The key to a module is to have the scalars (used in scalar multiplication of the vectors) as members of a ring instead of a field. A “$FG$-module” is a straightforward case of that. Since there is a field $F$ involved, a possible point of confusion is whether the scalars are members of the field $F$ now, instead of coming from a ring. But it is $FG$, a group ring, that is the set in which the scalars are members.

So to summarize: a $FG$-module is a module where the scalars are members of a group ring $FG$. The group ring $FG$ is defined and its module action is defined in the answer from Alex Ortiz above.