1. ## FG-modules

Hello, this is my first post.

I was wondering how, if we have a finite group $\displaystyle G$ and a field $\displaystyle F$ and an $\displaystyle FG$-module $\displaystyle FG$, how we would show there is no $\displaystyle FG$-submodule complementary to $\displaystyle V=\left\{\sum_{h\in G}x_h h\mid \sum_{h\in G} x_h=0\right\}$? Obviously a proof by contradiction but how would I start?

Thanks,

John

2. Originally Posted by JohnnyFrankston
Hello, this is my first post.

I was wondering how, if we have a finite group $\displaystyle G$ and a field $\displaystyle F$ and an $\displaystyle FG$-module $\displaystyle FG$, how we would show there is no $\displaystyle FG$-submodule complementary to $\displaystyle V=\left\{\sum_{h\in G}x_h h\mid \sum_{h\in G} x_g=0\right\}$? Obviously a proof by contradiction but how would I start?

Thanks,

John
this is certainly not true! for example if F is a field of characteristic 0, then FG becomes a semisimple algebra (Maschke's theorem) and thus every ideal (FG submodule) of FG has a complement.

3. Originally Posted by NonCommAlg
this is certainly not true! for example if F is a field of characteristic 0, then FG becomes a semisimple algebra (Maschke's theorem) and thus every ideal (FG submodule) of FG has a complement.
Oh of course, I meant to say that $\displaystyle |G|= 0\in F$. Otherwise yes maschke comes to the rescue haha

4. Originally Posted by JohnnyFrankston
Oh of course, I meant to say that $\displaystyle |G|= 0\in F$. Otherwise yes maschke comes to the rescue haha
suppose $\displaystyle FG=V \oplus W$ and let $\displaystyle e=\sum_{g \in G} g.$ see that $\displaystyle ge=e$ for all $\displaystyle g \in G$ and thus $\displaystyle ve=0,$ for all $\displaystyle v \in V.$ we have $\displaystyle 1=v+w,$ for some $\displaystyle v \in V, \ w \in W$ and so $\displaystyle e=we.$ now suppose that $\displaystyle e \in V.$

then since $\displaystyle V,W$ are $\displaystyle FG$ submodules, we'll have $\displaystyle e=we \in V \cap W = \{0 \}, \color{red}^*$ which is nonsense. hence $\displaystyle e \notin V,$ which means that $\displaystyle |G|,$ as an element of $\displaystyle F,$ is non-zero. Q.E.D.

$\displaystyle \color{red}*$: note that $\displaystyle V$ is an ideal of $\displaystyle FG,$ i.e. it's both left and right $\displaystyle FG$ sumodule. so if we assume that $\displaystyle W$ is a right $\displaystyle FG$ submodule and $\displaystyle e \in V,$ then $\displaystyle we \in V \cap W = \{ 0 \}.$

5. Originally Posted by NonCommAlg
suppose $\displaystyle FG=V \oplus W$ and let $\displaystyle e=\sum_{g \in G} g.$ see that $\displaystyle ge=e$ for all $\displaystyle g \in G$ and thus $\displaystyle ve=0,$ for all $\displaystyle v \in V.$ we have $\displaystyle 1=v+w,$ for some $\displaystyle v \in V, \ w \in W$ and so $\displaystyle e=we.$ now suppose that $\displaystyle e \in V.$

then since $\displaystyle V,W$ are $\displaystyle FG$ submodules, we'll have $\displaystyle e=we \in V \cap W = \{0 \}, \color{red}^*$ which is nonsense. hence $\displaystyle e \notin V,$ which means that $\displaystyle |G|,$ as an element of $\displaystyle F,$ is non-zero. Q.E.D.

$\displaystyle \color{red}*$: note that $\displaystyle V$ is an ideal of $\displaystyle FG,$ i.e. it's both left and right $\displaystyle FG$ sumodule. so if we assume that $\displaystyle W$ is a right $\displaystyle FG$ submodule and $\displaystyle e \in V,$ then $\displaystyle we \in V \cap W = \{ 0 \}.$
Thanks a lot for that - the two bits I don't quite understand I've highlighted in red in the quote above.

Essentially I don't understand why $\displaystyle ge=e$ for all $\displaystyle g \in G$ implies $\displaystyle ve=0$ for all $\displaystyle v\in V$, nor do I understand why $\displaystyle e\not\in V$ implies anything about the order of the group.

6. Originally Posted by JohnnyFrankston
Thanks a lot for that - the two bits I don't quite understand I've highlighted in red in the quote above.

Essentially I don't understand why $\displaystyle ge=e$ for all $\displaystyle g \in G$ implies $\displaystyle ve=0$ for all $\displaystyle v\in V$, nor do I understand why $\displaystyle e\not\in V$ implies anything about the order of the group.
if $\displaystyle v=\sum_{g \in G}x_gg \in V,$ then $\displaystyle \sum_{g \in G}x_g = 0$ and thus $\displaystyle ve=\sum_{g \in G} x_g ge=\sum_{g \in G}x_ge = \left(\sum_{g \in G}x_g \right)e = 0.$

we have $\displaystyle e=\sum_{g \in G}x_g g,$ where $\displaystyle x_g = 1,$ for all $\displaystyle g \in G.$ now $\displaystyle e \notin V$ means that $\displaystyle |G|=\sum_{g \in G}1 =\sum_{g \in G} x_g \neq 0.$ (see the definition of $\displaystyle V$ again!)