# Prove the existence of homomorphism.

I am trying to answer the following question. Is there any group homomorphsim $$\phi: D_4 \rightarrow S_5$$?

• (i) Is there a homomorphsim $D_4 \to S_4$? (ii) Is there a homomorphsim $S_4 \to S_5$? – copper.hat Dec 3 '18 at 21:44
• Given any two groups there's always a homomorphism $\phi:G\to H$. It might not be a very interesting homomorphism though. – Lord Shark the Unknown Dec 3 '18 at 22:14
• I mean a non-trivial homomorphism... – Felagamat Dec 3 '18 at 22:22

The subgroup $$\{e, (12)(34), (13), (13)(24), (14)(23), (24), (1234), (1432)\}$$ of $$S_4$$ is isomorphic to $$D_4$$. Hence we have injective group homomorphisms $$D_4\hookrightarrow S_4\hookrightarrow S_5.$$ Actually, all three Sylow-2-subgroups of $$S_4$$ are isomorphic to $$D_4$$.
• Perfect. But I see that there is a recipe to define a homomorphism $:D_n \rightarrow S_n$, for instance $\phi (\tau) =(1 2 3 4), \phi(\sigma)=(2 4)$. But is really enough to check that $\phi(\sigma \tau) =\phi (\tau^{-1} \sigma),$ to obtain the result? – Felagamat Dec 3 '18 at 22:21
• You just have to show that $(1234)$ and $(24)$ generated $D_4$. A homomorphism is determined by the images of the generators. – Dietrich Burde Dec 4 '18 at 9:11