I am trying to answer the following question. Is there any group homomorphsim $\phi: D_4 \rightarrow S_5$?
2026-03-26 01:14:32.1774487672
Prove the existence of homomorphism.
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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 $D_4$.