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

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

  • $\begingroup$ 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? $\endgroup$ – Felagamat Dec 3 '18 at 22:21
  • $\begingroup$ You just have to show that $(1234)$ and $(24)$ generated $D_4$. A homomorphism is determined by the images of the generators. $\endgroup$ – Dietrich Burde Dec 4 '18 at 9:11

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