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In the group $S_n$ consisting of the set of all bijections from $\{1,2,...,n\}$ to $\{1,2,...,n\}$ with a binary operation $◦$ denoting composition of functions. I am asked to find cyclic subgroups of $S_4$ with 2,3, and 4 elements. For a cyclic subgroup with 2 elements I have $\{(1,3),(2,4),(3,1),(4,2)\}$, but am looking for help in finding cyclic subgroups of larger sizes.

Daniel Jago
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    Your notation is very puzzling. (i) A subgroup has to contain the identity element, and I don't see it in the strange set you write down. (ii) As you want a subgroup of order 2 there's got to be one other element, of order 2. Did you intend the map that swaps 1 and 3, and swaps 2 and 4? If so we'd usually write it $\begin{pmatrix} 1& 2 & 3& 4\3 & 4 & 1 & 2\end{pmatrix}$ or just $(13)(24)$. – ancient mathematician May 25 '21 at 06:59
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    did you mean to say that each of the transpositions when considered separately along with the identity form a cyclic sub-group of size two? – C Squared May 25 '21 at 07:00
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    You may need to improve your notation, which is confusing. Also to specify a subgroup you need to identify all the elements. – Mark Bennet May 25 '21 at 07:00
  • Consider composing cycles of different sizes with themselves. For example, compose $(1,2,3)$ with itself and $(1,2,3,4)$ with itself to see what cyclic sub-groups those elements generate – C Squared May 25 '21 at 07:12
  • See here for the answer. There are $17$ cyclic subgroups of $S_4$. See the homework solution. – Dietrich Burde May 25 '21 at 12:16

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