Let $G$ be a subgroup of $GL_n(\mathbb K)$ and $H$ a subgroup of $GL_m(\mathbb K)$ where $\mathbb K \in \{\mathbb R, \mathbb C, \mathbb H\}$. I want to prove that there exists a subgroup of $GL_{n+m}(\mathbb K)$ isomorphic to $G \times H$.
Can you tell me if this is right: For $g \in G, h \in H$ let $M$ be the block diagonal matrix with diagonal blocks $g$ and $h$. Then the set of all such block diagonal matrices is a subgroup of $GL_{n+m}$ and isomorphic to $G \times H$. This is closed with respect to multiplication because of how multiplication of block diagonal matrices is defined.
Thanks.
You have the right idea, but you might want to be more explicit about proving closure. You can use a subgroup criterion here:
$$\left( \begin{array}{cc} g_1 & 0 \\ 0 & h_1 \end{array} \right) \left( \begin{array}{cc} g_2 & 0 \\ 0 & h_2 \end{array} \right)^{-1} = \left( \begin{array}{cc} g_1 g_2^{-1} & 0 \\ 0 & h_1 h_2^{-1} \end{array} \right)$$
Also, you need to prove this is isomorphic to $G \times H$, so use the fact that if $G' \cap H' = \{ I \}$ and if $G',H'$ are normal subgroups of $G'H'$, then $G' \times H' \simeq G'H'$.