Question

1.

Let $G$ be a $p$-solvable group and $V$ be a finite dimensional faithful $kG$-module, where the characteristic of $k$ is $p$. But $V$ is not a semisimple $kG$-module. For every $n\geq 0$, we define $End_{k}^{0}(V)=V$ and $End_{k}^{n}(V)=End_{k}(End_{k}^{n-1}(V))$ when $n>0$.

Let $L$ be a subgroup of $G$ such that $End_{k}^{n}(V)$ is a semisimple $kL$-module for every $n\geq 0$.

Is $L$ a $p'$-subgroup?

2.

Let $V$ be a 2-dimensional vector space over a field $k$ of characteristic $p$ and $G$ be a $p$-solvable subgroup of $GL(V)$, where $p$ is a prime number larger than than 5. But $V$ is not a semisimple $kG$-module.

Let $L$ be a subgroup of $G$ such that $V$ is a semisimple $kL$-module.

Is $L$ a $p'$-subgroup?

Answer 1

The answer to the second question is yes: As $G$ is reducible, this is true for $L$ even more. As $V$ is a semisimple $L$-module, $V$ is a sum of two $1$-dimensional $L$-modules. So $L$ is conjugate to a group of diagonal matrices. But elements of finite multiplicative order in $k$ have $p'$-order.