Subset Product of Normal Subgroups with Trivial Intersection
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Let $H, K$ be normal subgroups of $G$.
Let $H \cap K = e$.
Then $H K$ is isomorphic to $H \times K$ where:
- $H K$ denotes the subset product of $H$ and $K$
- $H \times K$ denotes the direct product of $H$ and $K$.
Let $G' = H K$.
That is $G'$ is itself a group.
The result follows by definition of the internal group direct product.