Showing two groups are not isomorphic using the order of their elements.

Question

I am trying to solve this question:

"$\text{Prove that no two of the groups } C_2 \times C_2 \times C_2 , C_2 \times C_4 \text{ and } C_8 \text{ are isomorphic.} $"

I understand that to show they are not isomorphic I need to show that there is a different number of elements of order $ n $ for some $ n\in \mathbb{N} $. But how do I show this?

I am confused as to how you determine the order of each element in a cyclic group. I know that there is an element of order 4 in $ C_2 \times C_4 $ because it is the lowest common multiple of 2 and 4, however how do I know if there is an element of order 4 in $ C_8 $ for example?

Any help would be much appreciated.

Answer 1

Hint: What is the highest element order in each group?