Is this true $(a*b)^q =(c*d)^q \implies a*b=c*d?$

Question

Is this true $(a*b)^q =(c*d)^q \implies a*b=c*d$?

Here $a,b,c,d \in G$, $(G,*)$ forms a group, and $q$ is an integer

Answer 1

No, consider the cyclic group $\mathbb{Z} / q\mathbb{Z}$. Then the left-hand side is always satisfied whereas the right-hand side in general is not.

Answer 2

No, not even for infinite groups.

Let $a=b\neq c=d, q=2$. Consider the group given by the presentation $$\langle a, d\mid a^4, d^4\rangle.$$ This is the free product of two different presentations of $\Bbb Z/4\Bbb Z$.

We have $a^4=d^4$ but $a^2\neq d^2$.

Answer 3

Choose two units $a,b\in\mathbb Z_p$, then $a\neq b$ but $$a^{p-1}=b^{p-1}=1$$ by Fermat.