I'm looking for an answer to the following problem, any responses would be greatly appreciated as I think it's quite complicated!.
Let $U=\mathbb{F}_{q}^{m}$ and $W=\mathbb{F}_{q}^{n}$ be two vector spaces of dimension $m$ and $n$ over $\mathbb{F}_{q}$, respectively. Let $V_{1}=U \oplus W=\mathbb{F}_{q}^{m+n}$ and $V_{2}=U \otimes V=\mathbb{F}_{q}^{mn}$. Let $G$ be the subgroup of $GL(V_{1})=GL(m+n,\mathbb{F}_{q})$ that fixes $U$. Let $H$ be the group $(\mathbb{F}_{q}^{+})^{mn}:(GL(m,\mathbb{F}_{q})\times GL(n,\mathbb{F}_{q}))$ induced by the natural action of $GL(m,\mathbb{F}_{q})\times GL(n,\mathbb{F}_{q}))$ on the tensor product $V_{2}$, where $:$ denotes the semi-direct product. Prove that G and H are isomorphic.