Is it true that in any group $|ab| = |ba|$.

Question

Is it true that in any group order of the element $ab$ and $ba$ are same? Could anybody help me to clear my doubt?

Thanks

Answer 1

What have you tried?

Here's a hint: $|g|=|h^{-1}gh|$. (Proof: If $g^n=e$, then $(h^{-1}gh)^n=h^{-1}g^nh=h^{-1}eh=e$.)

Answer 2

Assume that the order of $ab$ is $n$. We then have that $(ab)^n = e$. Therefore $(ab)^n = a(ba)^{n−1}b = e$. Multiply both sides of this last equation on the right by $a$ and you get $a(ba)^n = a$. Multiply both sides on the left by $a{−1}$ and you get $(ba)^n = e$. The same argument shows that if $(ba)^n = e$ then $(ab)^n = e$. (just reverse the roles of a and b). Therefore if either of these products has order n then so does the other.