What are all of the possible orders of $K$? Justify your answer.

Question

Let $K\le A_5$. Assume that $K$ is cyclic. What are all of the possible orders of $K$? Justify your answer.

So I know that $|K|\in\{1,2,3,5\}$ but I'm not sure how to justify it.

Answer 1

Suppose $K\leqslant A_5$; with $K$ cyclic. Then $K$ is generated by one element, $\sigma\in A_5$, so $K=\langle \sigma\rangle$. Now, it remains to determine the order of $\sigma$. Now, what are the possible cycle types of $\sigma$? Why does this give you the answer? For example, $\sigma$ can be the identity, so $|K|=1$. It can also be a product of disjoint transpositions, and $|K|=2$.

SPOILER

Remember that the cycle type of an element in $S_5$ can be $(1),(12),(123),(1234)(12345)$,$(123)(45),(12)(34)$ but only $(1);(123),(12345),(12)(34)$ are in $A_5$. Thus $\sigma$ has cycle type $(1),(123),(12345),(12)(34)$. These elements have order $1,3,5,2$ respectively, and those are the possible orders.