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I'm just beginning my course in algebra. I've been reading Milne, Group Theory ( http://www.jmilne.org/math/CourseNotes/GT310.pdf page 29).

I've found there a very nice proof of the fact that given two elements in a finite group, we cannot really say very much about their product's order. However, there are some things about the proof I do not quite understand. Namely - the first paragraph. What are the images of elements in \(\displaystyle SL_2(\mathbb{F}_q)/ \{+-I\}\) and why do we divide the orders of \(\displaystyle a, \ b, \ c\) by \(\displaystyle 2\)?

Is it because the centre(\(\displaystyle \{+-I\} \)) has order \(\displaystyle 2\) and thus by Lagrange's theorem, the order of the quotient group must be two times smaller?

I would really appreciate a thorough explanation. Maybe you know a simpler proof of the fact (about the order of product of elements)?

Thank you.