Answer to Question #186471 in Abstract Algebra for rdh

Question

give an example of a nontrivial homomorphism or explain why none exists. φ:S₃ → S₄

Answer 1

Note that by definition, the trivial homomorphism is the map $f: S_3\to S_4,\ f(s)=(1)$ for all $s\in S_3.$ Each permutation $s$ of the symmetric group $S_3$ on the set $\{1,2,3\}$ can be identified with the permutation of the symmetric group $S_4$ on the set $\{1,2,3,4\}$ by putting $s(4)=4.$ It follows that for the map $\varphi:S_3\to S_4,\ \varphi(s)=s,$ we have that $\varphi(s\circ t)=s\circ t=\varphi(s)\circ \varphi( t)$, and hence the map $\varphi$ is a nontrivial homomorphism.