Question

If (a,b) E AxA, where A is a group, then o((a,b))=o(a)o(b)

Answer 1

Let $A$ be a group. If $o(a)=o(b)=2$, then $(a,b)(a,b)=(a^2,b^2)=(1_A,1_A)$, and hence $o((a,b))=2$. It follows that in general case it is not true that $o((a,b))=o(a)o(b).$