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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?

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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 $End_{k}^{n}(V)=End_{k}^{n-1}(V)$ 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$is a prime number larger than than 5. Let$L$be a subgroup of$G$such that$V$is a semisimple$kL$-module. Is$L$a$p'\$-subgroup?

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