Does the existence of a homomorphism and a bijection between two groups imply the existence of an isomorphism?
2026-03-31 17:47:46.1774979266
Question about Homomorphism and Isomorphism
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The answer remains no even if we require a bijection and a monomorphism (injective homomorphism): Consider the multiplicative group $\mathbb C^\ast$ of complex numbers and $\mathbb R^\ast$ of real numbers respectively. They satisfies the stronger requirement by the usual embedding, but are non-isomorphic as they have a different number of roots of unity.