G is an abelian group.
a,b are in G
order of a = m
order of b = n
what is this order of ab ?
Struggling with this. I suspect it is lcm(m,n) but unable to prove it.
Any help/pointers please?
G is an abelian group.
a,b are in G
order of a = m
order of b = n
what is this order of ab ?
Struggling with this. I suspect it is lcm(m,n) but unable to prove it.
Any help/pointers please?
it's not necessarily lcm(m,n). the only thing we can say is that $\displaystyle o(ab)$ divides lcm(m,n). the point is that $\displaystyle o(ab)$ doesn't only depend on $\displaystyle o(a)$ and $\displaystyle o(b)$. it also depends on the relationship between $\displaystyle a$
and $\displaystyle b$. for example if $\displaystyle b=a^{-1},$ then $\displaystyle o(a)=o(b)=m$ but $\displaystyle o(ab)=1.$