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

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.

• Note: $C_2\times C_2\times C_2$ has order $8$ (not $6$). All three groups have the same order. – paw88789 Mar 4 '15 at 22:11
• Oh of course. Because it wouldn't be 2x2x2, but (2x2)x2? – kw3rti Mar 4 '15 at 22:13
• Either of those expressions is correct for the order. Both work out to $8$. – paw88789 Mar 4 '15 at 22:14
• Yeah sorry, don't know how I made that mistake! – kw3rti Mar 4 '15 at 22:19

• The highest element order is not $8$ for each group. – paw88789 Mar 4 '15 at 22:59