[Math] Suppose that a is a group element and $a^6 = e$. What are the possibilities for $|a|$

Question

Suppose that $a$ is a group element and $a^6 = e$. What are the possibilities for $|a|$? (Gallian, Contemporary Abstract Algebra, Exercise 18, Chapter 3.)

I just started looking at Abstract Algebra again and I was stuck on this question. It will probably be extremely simple for all of you but I didn't know what to do.

I tried doing regular operations like those found in arithmetic but obviously, that is one of the reasons why Abstract Algebra is so difficult.

Answer 1

Wouldn’t it just be the divisors of 6?

For instance, if $|a| = 2$, then $a^6$ would be $e$, in virtue of:

$a^6 = (a^2.a^2).a^2 = (e.e).e = e.e = e$