Subgroup of $GL_{n+m}(\mathbb K)$

Question

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.

Answer 1

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 $G' \times H' \simeq G'H'$.