## Group action and semidirect product

Let $m,n \in \mathbb N$, $k$ a field, $X=(k^{n\times m},+)$, and we consider the groups $GL(n,k)$ and $GL(m,k)$. Let $K:=GL(n,k) \times GL(m,k)$. We define \begin{align*} K\times X &\to X\\ ((

mathematics abstract-algebra group-theory matrices morphism semidirect-product