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Ok so the Fundamental Homomorphism Theorem (or First Group Isomorphism Theorem) states that if θ : G → H and ker (θ) = K, then the quotient group G/K is isomorphic to H.

I know that θ : G → G has to be a surjective homomorphism with kernel {e}, but I'm not sure how to prove this or if it is automatically assumed. Help?

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Hint: Is $x\mapsto x$ a homomorphism?

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  • $\begingroup$ Ok that makes sense. So then will {e} be the kernel since I don't have any other known elements in G? $\endgroup$ – Amani Borders Apr 19 '15 at 18:10

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