Proof verification about inexistence of a surjective homomorphism between $C_8\times C_2 \to C_4 \times C_4$

Asked Jul 27 '21 at 11:17

Active Jul 27 '21 at 11:17

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Q. Prove that there does not exist a subjective homomorphism $C_8\times C_2 \to C_4 \times C_4$

My attempt: FSOC, assume there is such a homomorphism. As $|C_8\times C_2| = |C_4 \times C_4|$, the homomorphism must be bijective and hence an isomorphism. This means the two groups are isomorphic. But this is a contradiction as the former has an element of order 8 while the latter doesn't.

Thanks in advance for your help! :)

asked Jul 27 '21 at 11:17

Mathaddict

Yeah, the linked question answers my doubt! thanks! – Mathaddict Jul 27 '21 at 11:30
Should I delete my question? – Mathaddict Jul 27 '21 at 11:31
1

You don't have to, @mathematics_2001 – Shaun Jul 27 '21 at 13:42

Proof verification about inexistence of a surjective homomorphism between $C_8\times C_2 \to C_4 \times C_4$

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