It is not true for $G/H\times H\simeq G$ to hold for a subgroup $H\leq G$ when we talk about group isomorphisms.
However, what about topological groups and a homeomorphism in the above? (without being a group morphism)
2026-03-25 01:31:12.1774402272
Is $G/H\times H\simeq G$ topologically?
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No. Take $G=\mathbb{R}$ and $H = \mathbb{Z}$. If you consider the quotient $G/H$ as the group version correctly topologized, then $\mathbb{R}/\mathbb{Z} \times \mathbb{Z}$ is not homeo to $\mathbb{R}$. For one, the former isn't even connected (it's a stack of circles).
I'm also sure there are simpler examples than this one.