Let $G$ be the cyclic group of order 2 and let $V$ be the regular $\mathbb{F}G$-module of $G$. Determine the $\mathbb{F}G$-submodules of $V$ where $\mathbb{F}$ has each of zero or positive characteristic.
Remark: The group algebra $\mathbb{F}G$ and the trivial subspace $\{0\}$ are clearly two submodules of $\mathbb{F}G$. Thus one needs to find (if there's any) all 1-dimensional submodules.
Let $\lambda,\mu \in \mathbb{F}$ not both zero such that $W:=\langle \lambda a+ \mu 1 \rangle$ is a submodule of $\mathbb{F}G$. By $(\lambda a+\mu 1)a=\mu a+\lambda \in W$ to get the exisistence of a $k \in \mathbb{F}$ such that $\mu a+\lambda 1=k(\lambda a+\mu 1) $. Obseve that $\mu, \lambda,k$ are all non-zero (as one is zero would then imply the other two are zero), thus $k=\frac{\lambda}{\mu}=\frac{\mu}{\lambda}$ imples $k^2=\mu^2$, so either $\lambda=\mu$ or $\lambda=-\mu$
Now, if $char(\mathbb{F})=0$ then $W$ can be $\langle a+1 \rangle$ or $\langle a-1 \rangle$.
On the other hand, if $char(\mathbb{F})=p$ for some prime $p$, then $\lambda=tp-\mu$ or $\lambda=tp+\mu$ for any $t \in \mathbb{Z}$, the correspoding $W$ is thus $\langle \lambda a+(tp-\lambda) 1\rangle$ or $\langle a-(tp-\lambda) 1 \rangle$.
Does above imply the $\mathbb{F}G$ submodules are independent of $char(\mathbb{F})$? Since in the $char(\mathbb{F})>0$ case one has $tp=0,\forall k \in \mathbb{Z}$ ?