Given a group $G$, $a,b\in G$, $ab=ba$, $o(a)=n$, $o(b)=m$.
If $\gcd(n,m)\ne 1$, and $(a)\cap (b) = \{e\}$, prove that $o(ab)=\operatorname{lcm}(n,m)$.
P.S. $(a)$ denotes the cyclic group generated by the element $a$.
I’m a little confused... Aren’t the conditions after “if” contradictory?
Thanks for your help down there. But...can someone show me how to prove this?
Any help would be appreciated.