Suppose that there is a group $G$ and an odd prime $p$. Let $M$ be a $\mathbb{F}_p[G]$ module.
Is it true that $M$ cannot be irreducible but just indecomposible?
Thank you
Suppose that there is a group $G$ and an odd prime $p$. Let $M$ be a $\mathbb{F}_p[G]$ module.
Is it true that $M$ cannot be irreducible but just indecomposible?
Thank you
It depends on $M$, of course.
For example, suppose $G=C_p$ is the ciclic group of order $p$, let $\sigma\in C_p$ be a generator, and let $M=\mathbb F_p\oplus\mathbb F_p$ be the $\mathbb F_p[G]$-module on which $\sigma$ acts as the matrix $\begin{pmatrix}1&1\\0&1\end{pmatrix}$. You should not have problems showing that this is a non-irreducible indecomposable module.
It also depends on $p.$ If $F$ is a field of prime characteristic $p$ which does not divide the order of the finite group $G,$ then all indecomposable $FG$-modules are irreducible. On the other hand, if the prime $p = {\rm char}F$ divides $|G|,$ then there is always an indecomposable $FG$-module which is not irreducible ( for otherwise the group algebra $FG$ would be semisimple, and hence would have trivial Jacobson radical. But the sum of the group elements is a non-zero central nilpotent element of the center of $FG),$ so lies in the Jacobson radical of $FG.$