Find all of the elements of a subgroup generated by a $3$-cycle notation Question: Let $H$ be the subgroup of $A_n$ generated by $(123)$. Write down all elements of $H$. Here is my attempt: $H\le \langle 123 \rangle$  $H$=$\langle a \rangle$ where the order of $a$=$n$ where $\langle a \rangle$=$\langle e,a,a^{2},....,a^{n-1} \rangle$ since $\langle 123 \rangle$ has order $3$, $\langle 123 \rangle$=$\langle (e),(123),(23)(31) \rangle$ I computed $(123)^{2}$ and $(123)^{3}$ since I had to order $3$ to come up with my answer for $\langle 123 \rangle$ Did I do this right?

Yes, your answer is correct. $H = \langle (123) \rangle = \{e, (123), (132) \}$.
