Let G = <x> be a cyclic group of order 24. List all the elements in G that are of order 4.

Question

Let G = be a cyclic group of order 24. List all the elements in G that are of order 4.

The only relevant theorems in my book that I can find is:

The order of a group G, denoted by |G|, is the number of elements in G.

If G = , then |G| = o(x).

I'm not sure where to go with this. From the first theorem, I'm led to believe that the elements would just be $x^1, x^2, x^3, x^4$. Is this correct?

Answer 1

Hint: Here are the relevant elementary facts:

• All elements of $G=\langle x \rangle$ are of the form $x^k$ with $k\in \{0,1,\dots,23\}$.

• If $o(x^k)=4$, then $(x^k)^4=1$.

• If $x^m=1$, then $m$ is a multiple of $o(x)=24$.