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Let $\phi:G\to H$ be a homomorphism of groups. Under what conditions does $\frac{G}{ker (\phi)} \simeq im(\phi)$ imply there exists an isomorphism
$G\simeq im(\phi) \oplus ker(\phi)$, and why?

The answers in the possible duplicate question are not sufficient for me, as I do not have an understanding of category theory or exact sequences. I'm looking for a detailed answer that makes use of basic group theory alone. In particular I am interested in knowing if $G$ or $H$ being abelian makes any difference.

vkan
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  • I have looked at the answers to the similar questions mentioned, but I do not understand the answers. Need an answer that does not refer to exact sequences or category theory. – vkan Jan 24 '18 at 17:50
  • $G$ or $H$ being abelian doesn't help. – Randall Jan 24 '18 at 19:08
  • There is no need to learn about category theory, but you definitely need to learn about exact sequences. – Derek Holt Jan 24 '18 at 20:09

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