Ok, so there is this problem that quite confuses me because the example seems too easy (at least thats what I think):

Find an example of a group homomorphism $\Phi: S_3 \rightarrow S_4$ that is injective.

I think the example is just $\Phi(\sigma)=\sigma$, this is the only example I can think about but I am not sure if it is correct. Can anyone tell me if this example is valid? If not what other examples could there be?

  • $\begingroup$ If $\sigma\in S_3$, then how can $\sigma$ be in $S_4$? $\endgroup$ – Lord Shark the Unknown Dec 9 '17 at 8:06
  • $\begingroup$ Technically $\sigma \in S_3$ is also in $S_4$ no? $\endgroup$ – Aurora Borealis Dec 9 '17 at 8:46

Your example is (almost) right. If you concretely take $S_n$ as the permutations of $\{1,2,...,n\}$ then any $\sigma \in S_3$ can be mapped to $g(\sigma) \in S_4$ by having $g(\sigma)(4)=4,$ while for $k=1,2,3$ put $g(\sigma)(k)=\sigma(k).$

  • $\begingroup$ Wait so I dont get it, is it correct to say it is $\Phi(\sigma)=\sigma$ is indeed an injective homomorphism, for all $\sigma \in S_3?$ $\endgroup$ – Aurora Borealis Dec 9 '17 at 8:47
  • $\begingroup$ @AuroraBorealis Not technically, since you want $\Phi(\sigma)$ to be in $S_4.$ And if $\sigma$ is in $S_3$ that's not in $S_4,$ unless in a loose sense in which you always map $4$ to itself. $\endgroup$ – coffeemath Dec 9 '17 at 8:50

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