Trying to solve this question for the following groups and not quite sure how to do it.
Given:
i) $G1 = ℤ_{11}, G2 = ℤ_{12}$ | order 4
ii) $G1 = U(19), G2 = U(19)$ | order 6
iii) $G1 = D_4, G2 =Q_8$ | order 4
I want to find the number of elements of the given order in $G1×G2$.
Now using what I know, trying for the first one, $|(a,b)| = lcm(|a|, |b|) = 4 \implies ... ?$
Now I thought it would imply $(|a|, |b|) = (4,1), (4,2)$ but that leads me to 4 which isn't the correct answer. The notes I'm using are a bit confusing so I was hoping for clarification and help with regards to answering this. Thanks!