Let $\varphi:Q_8 \to S_4$ where $\varphi(i)=(ab)$ and $\varphi(j)= (cd)$.Then is this a homomorphism? I think it is because all the conditions for homomorphisms are satisfied. Now one more homomorphism is from $Q_8$ to the klein-four group in $S_4$.The trivial homomorphism is one more case.Are these all?
2026-04-03 02:00:26.1775181626
Prove whether it is a homomorphism.
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