counterexample showing that Maschke's Theorem does not hold if characteristic divides group order

Question

I am taking a graduate Algebra course, and we were given the following example to see that Maschke's Theorem does not hold if the characteristic of the field F does divide the order of G:

Let $F = \mathbb{F}_2$, $G = \mathbb{Z}/\mathbb{Z}_2$ and $V=FG$. Then show that $W = span\{0+1\}$ is not complemented. Hint: Show that there is no other submodule of dimension 1.

Now I think that I got something basic wrong because, as I understood it, $span\{0+1\}=G*\{0+1\}=\{0*0+0*1; 1*0+1*1\}$=$\{0+0;0+1\}$ with $*$ being the action of G on V. But then why is for example $span\{1\}$ not a submodule of dimension 1?

Can someone tell me where my error is? Thank you very much!

Answer 1

if the group is $\{1,a\}$ with $a^2=1$ then over $F_2$ $$ (1+a)^2= 1 + 2a+a^2 \equiv_2 0 $$

since the element $1+a$ is nilpotent the group algebra is not semisimple

Answer 2

In fact the conclusion of Maschke's theorem never holds in the modular case, that is, suppose that ${\rm char}\, k =p\mid G$. Consider the element $\eta=\sum_{g\in G}g\in kG$. Then $g\eta=\eta$ for every $g\in G$ and then the ideal generated by $\eta$ is the same as the $k$-subspace generated by $\eta$, i.e. $\langle \eta\rangle =(\eta)$. Moreover $\eta^2=|G|\eta=0$. It follows $kG$ contains nilpotent ideals, so it is not semisimple -- recall that an algebra is semisimple iff it is artinian and contains no nonzero nilpotent ideals.