Using Cayley’s theorem, I want to construct a transitive subgroup of $S_6$ which is isomorphic to $S_3$. I've been struggling with group theory rather a lot, and I find this particular problem to be very difficult.
Cayley's Theorem states that: "Every group $G$ is isomorphic to a subgroup of $S_G$". What exactly does this imply for the problem that I want to solve? I know what a transitive group is, but I don't know how Cayley's Theorem comes into this problem.
On a related note, does one develop an intuition for group theory with enough practice? I've had enormous issues with the general abstractions so far. Thanks in advance.


It is not a bad idea to make this abstract algebra less abstract.
The subgroup of $S_6$ generated by $\sigma=(1\,2\,3)(4\,5\,6)$ and $\tau=(1\,4)(2\,5)(3\,6)$ is the symmetry group of a triangular prism, and it is pretty obviously transitive and isomorphic to $S_3$.

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  • $\begingroup$ You are right, the graphical depiction does make it rather obvious. I guess that might be an approach that I should try in the future. Thank you. The actual subgroup would have to include the unit element (1), though, right? $\endgroup$ – user569579 Dec 6 '18 at 10:05
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    $\begingroup$ @user569579: $\sigma^3=\tau^2=(1)(2)(3)(4)(5)(6)$. $\endgroup$ – Jack D'Aurizio Dec 6 '18 at 14:22

Hint: There are six elements of $S_3$, and the elements in $S_3$ canonically act on these by ...

  • $\begingroup$ I'm sorry, but what do you mean with "canonically acting on"? I am not familiar with that terminology. $\endgroup$ – user569579 Dec 5 '18 at 22:39
  • $\begingroup$ @user569579 You can swap "canonically" with "naturally". You want a subgroup of $S_6$, so you need a set of six elements you can act on (permute). $S_3$ has six elements. Now, here it may be difficult to keep everything straight: You have the set $S_3$, and the group $S_3$ acting on that set in some way. What is the most natural way you can think of to combine an element from the group $S_3$ with an element from the set $S_3$ to make an element of the set $S_3$? $\endgroup$ – Arthur Dec 5 '18 at 22:50
  • $\begingroup$ I would say by acting on it with the unit element $e$, so that the element remains the same. That way, all the elements are preserved. $\endgroup$ – user569579 Dec 5 '18 at 23:09
  • $\begingroup$ Yeah, but that's not transitive. In what other way can you combine two elements of $S_3$ to make a new element of $S_3$? $\endgroup$ – Arthur Dec 5 '18 at 23:21
  • $\begingroup$ I believe by acting on it with a cycle that has the same length as the element. Perhaps I'm just talking nonsense now, though... Like I said, I find this topic very difficult. $\endgroup$ – user569579 Dec 5 '18 at 23:37

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