Let $V$ be an elementary abelian $p$-group of size $p^n$. Let $G$ be a finite group with $V\unlhd G$ such that $G/V=H$ is simple (like $\operatorname{PSL}(m,q)$ with $q$ a power of $p$ or any other finite simple group of Lie type in characteristic $p$). Moreover, $V$ is a faithful irreducible $H$-module (equivalently, $V$ is a minimal normal subgroup of $G$ and $C\_G(V)=V$).


Ideally, I would like to show that $G$ must be the semidirect product $V\rtimes H$, or $|V|$ must be somewhat large compared to $|H|$. **A quick example:** when $p=2$ and $H=\operatorname{PSL}(n,2)$, so that $V$ is the natural $n$-dimemsional $\mathbb{F}\_2$-module for $H$, according to "U. Dempwolff, On the second cohomology of $\operatorname{GL}(n,2)$", $G$ must be the mentioned semidirect product.


Any suggestions and references are appreciated.



Extensions of a simple group by an elementary abelian $p$-group
I think this can often fail even for the non-trivial representation $V$ of smallest possible dimension.


For a reductive group $G$ over $\mathbf{Z}\_p$, the group $G(\mathbf{Z}/p^2 \mathbf{Z})$ will typically be a non-split extension of $G(\mathbf{F}\_p)$ by the adjoint representation $V$, and for various $G$ the adjoint representation $V$ will be the smallest non-trivial representation. For example, take $G = E\_8$.


An even simpler example is the group $\Gamma = \mathrm{PSL}\_2(\mathbf{F}\_p)$ for $p > 3$. Then the smallest non-trivial representation over $\mathbf{F}\_p$ will be the adjoint representation $V$ of dimension $3$, but $\mathrm{PSL}\_2(\mathbf{Z}/p^2 \mathbf{Z})$ gives a non-split extension of $\Gamma$ by $V$, since the lift of any unipotent element in $\Gamma$ will have order $p^2$.


