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I'm am working on this question:

Find a subgroup of the symmetric group $S_4$ that is isomorphic to $S_3$.

What I know (thus far):

Motivated from one of my past questions, I have that $$S_3 = \{(1),(12),(13),(23),(123),(132)\}$$ Thus, I can chose (with some relabeling) $$H_a = \{(1),(12),(13),(23),(123),(132)\}$$

as a subgroup of $S_4$. If this is correct, why so?

John W. Smith
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  • I think it would help to clarify your notation: since $S_3$ acts on three elements and $S_4$ on four elements, you could write $H = {(1)(2)(3)(4),(12)(3)(4), \ldots, (132)(4)}$. Then show an isomorphism between $H$ and $S_3 = {(1)(2)(3), (12)(3), \ldots, (132)}$. – Théophile Apr 26 '17 at 17:42
  • Ah yes. I will edit my question later this afternoon when time permits. – John W. Smith Apr 26 '17 at 17:57
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    To show the isomorphism, try sending a permutation $\sigma \in S_3$ to $\sigma(4) \in H$. – Théophile Apr 26 '17 at 18:09
  • Ah so as long as $4$ is fixed, I can send any permutation $\sigma$ from $S_3$ and it would be in $S_4$ correct? – John W. Smith Apr 26 '17 at 18:32
  • Yes, that's right. Of course, $4$ is arbitrary; you could fix any other element of ${1,2,3,4}$ to end up with a different subgroup isomorphic to $S_3$. – Théophile Apr 26 '17 at 18:36
  • I see. Well that clears my doubts then. – John W. Smith Apr 26 '17 at 18:49

1 Answers1

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Yes, it is correct. The idea behind that is the following.

The group $S_4$ is the group of permutations of $4$ objects, let's call them $\{1,2,3,4\}$. The subgroup you defined simply is the subgroup of $S_4$ given by permutations that don't move $4$. It should be obvious why it is isomorphic to $S_3$.

Daniel Robert-Nicoud
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  • Well as you said,we leave $4$ fixed. Since I'm fairly new to the concept of isomorphsms, how would I show that there is an isomorphism between $H$ and $S_3$, using Théophile's remarks? I don't see how this is obvious. I $do$ know that there must be a bijection of some sort and that the group operations are preserved. – John W. Smith Apr 26 '17 at 17:59
  • @i8Σπ_821 Just define a map $S_3\to S_4$ by sending $\sigma\in S_3$ to the element of $S_4$ keeping $4$ fixed and acting as $\sigma$ on ${1,2,3}$, and show that it is an injective group homomorphism. – Daniel Robert-Nicoud Apr 26 '17 at 20:15
  • That's what I had in mind and Théophile cleared it up as well. – John W. Smith Apr 26 '17 at 22:05