# questions about representation theory/structure theorem for finitely generated modules

I am reading a book in representation theory by James and Liebeck. They define an FG-module as: Let $V$ be a vector space over the field $F$ 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) $vg \in V$, (2) $v(gh) = (vg)h$, (3) $v1=v$, (4) $(\lambda v)g = \lambda(vg)$, (5) $(u+v)g =ug + vg$

They state that they use the letters $F$ and $G$ to indicate that $V$ is a vector space over $F$ and that the elements $g$ comes from $G$. I am asking for analogs to the definitions of a R-module, or any discussion, clarification or generalisation of the definition above. Also if $V$ is an FG-module as defined above then there is a basis for the module and hence it is finitely generated? In the following chapters the connections between FG-modules and group representations are explored, but I wonder if some of these results may hold for any finite generated module, what is the connection? Also I wonder if there is any representation theoretical approach to prove the Structure theorem for finitely generated modules over a principal ideal domain or any special case of this theorem, because I suspect there is... I am a totally beginner to representation theory and modules and group algebras so I am basically looking for a bigger picture to give things a little bit more connection and motivation.

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This is very vague. Can you ask a more precise question? – Qiaochu Yuan Sep 17 '12 at 22:59

There is the so called 'group algebra' $F[G]$ which is the $F$-vector space with formal basis $G$, this also becomes a ring by the multiplication of $G$. Since $F$ is commutative, it doesn't matter if it acts from left or right. Then, $FG$-modules as defined above straightly correspond to right $F[G]$ modules.

If $V$ has finite dimension, then it stays finitely generated as $F[G]$-module.

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As for an analogue of the structure theorem: This depends on what aspect of the theorem you are interested in.

If you are interested in the property that modules have a unique direct sum decomposition then this is the Theorem of Krull-Remak-Schmidt: Every noetherian and artinian module has a unique (up to permutation and isomorphism) decomposition into indecomposables.

If you really want to know how the indecomposables can look like you might be surprised to see that it is in most cases not possible to write down a complete list of all indecomposable modules. (Algebras are distinguished via their representation type) This happens even for quite innocently looking groups like $\mathbb{Z}/(3)\times \mathbb{Z}/(3)$ (if the field has characteristic $3$.

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