Any group which is of prime order is a cyclic group

Question

I don't know how to prove this:

Any group which is of prime order is a cyclic group.

What fact should I use to prove this?

Answer 1

Hint: Suppose $|G|=p$ where $p$ is a prime.

Suppose, $a\in G$. What can you say about $|a|$?

Note $|.|$ denotes order.

Answer 2

1. Take a nonzero element a in the group.

2. Consider the group generated by a, which is of course cyclic.

3. Try to recall what you can say about the order of a subgroup relative to the order of the group itself, and what this implies in your case.