What is the difference between an $FG$-module and a group algebra?

Question

I am beginning the study of representation theory, and am having trouble understanding the difference between an $FG$-module and a group algebra of $G$. This seems to be the only other question on this topic, but I don't think the answer is relevant to the definitions I have.

In the course I am taking, we have the following definition for an $FG$-module:

Let $V$ be a vector space over $F$ and let $G$ be a group. Then $V$ is an $FG$-module if a multiplication $vg (v \in V, g \in G)$ is defined, satisfying the following conditions for all $u, v \in V, \lambda \in F$ and $g, h \in G$ :

1. $v g \in V$;
2. $v(g h)=(v g) h$
3. $v 1=v$;
4. $(\lambda v) g=\lambda(v g)$;
5. $(u+v) g=u g+v g$.

Now, the definition of group algebra is given:

The vector space $F G$, with multiplication defined by $$\left(\sum_{g \in G} \lambda_g g\right)\left(\sum_{h \in G} \mu_h h\right) = \sum_{g, h \in G} \lambda_g \mu_h(g h)$$ $\left(\lambda_g, \mu_h \in F\right)$, is called the group algebra of $G$ over $F$.

In my mind these are essentially the same thing. Both are a "precise combination" of a field and the elements of a group such that multiplication is defined with closure, associativity, identity, scalars, and distributivity. I don't really see what makes them distinct.

I suppose there is some sort of "hierarchy" that goes $$\text{Group Algebra}\subset \text{FG-module}\subset\text{Vector space}?$$ Is my understanding correct? I feel pretty flaky about it.

Answer 1

From the comments:

If you know what a module over a ring is, and you know what an algebra over a field is, you must see that the two definitions are different. For example, you can take two elements in an algebra and multiply them to obtain a third one, but if you have a module then you cannot multiply two of its elements.

A module is like a vector space, just defined over a ring instead of a field. So first we define the group algebra $FG$ (if it's an algebra then it is in particular a ring), and then we consider modules over that ring. Now, your definition of an $FG$-module looks a bit different (more confusing), but it's equivalent to what I wrote. By your definition, the "scalar multiplication" is only defined by scalars from the group $G$. But it can be checked that this can be naturally extended to scalar multiplication with scalars from $FG$, and so we get a module over $FG$. (note that $G\subseteq FG$)

This is one of those situations where people will be confused at your confusion: an $\mathbb k$-algebra is something that behaves like the space of endomorphisms of a $\mathbb k$-vector space: $\operatorname{End}(V)$ is a vector space along with a product (composition of functions) which is $\mathbb k$-bilinear. Because algebras are modelled in this way, you can consider algebra maps $A\longrightarrow \operatorname{Env}(W)$ where $W$ is some vector space, and this $W$ is an $A$-module (i.e., a way of seeing $A$ as a collection of operators on $W$). They are very different things!