- #1

JonnyG

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Corollary 2.5.9: A homomorphism ##\varphi: G \rightarrow G'## is injective if and only if its kernel ##K## is the trivial subgroup ##\{1\}## of ##G##.Obviously Corollary 2.5.9 implies that the homomorphism is an isomorphism onto its image, but I do not see how it implies it's an isomorphism onto the co-domain.

Counter-example: Define ##f: \mathbb{R}^+ \rightarrow \mathbb{R}^{\times}## where ##\mathbb{R}^+## is the additive group of ##\mathbb{R}## and ##\mathbb{R}^\times## is the multiplicative group of ##\mathbb{R}##. Obviously ##f## is an isomorphism onto its image (the multiplicative group of positive real numbers), but it's clearly not a surjection.

Did Artin make a mistake or am I missing something painfully obvious?