Determine the cyclic subgroup of $(S_5,\circ)$, the symmetric group on five symbols, generated by the cycle $(1 , 3, 5 , 2, 4)$.

Question

Request vetting, as doing first time.

Let, $a= (1 , 3, 5 , 2, 4)$, then have $$a^2= (1 , 5, 4, 3, 2),$$ $$a^3= (1, 4, 2, 5, 3),$$ $$a^4= (1, 2, 3, 4, 5)=e.$$ So, get order of this cyclic subgroup (as here) $=4$, with elements:$\,\{e, a,a^2,a^3\}$.

Edit

The one line form was wrongly used to achieve squaring each time.

$$a^3= (1, 2, 3, 4, 5) = e,$$ $$a^4= (1, 4, 2, 5, 3).$$

Answer 1

You forgot the identity (in the original version of your question).

The cyclic subgroup contains all powers of the generator $a$, so that includes $a^0=e$.

Also,

$$\begin{align} a^3&=(13524)a^2\\ &=(13524)(15432)\\ &=(12345) \end{align}$$

and

$$\begin{align} a^4&=(13524)a^3\\ &=(13524)(12345)\\ &=(14253), \end{align}$$

so that $a^4(1)=4\neq 1$, meaning $a^4\neq e$.