In a Group $G$ if we have $a^n=b^n\neq e$ does it imply that $a=b$?

Question

Given $G$ a group and $a,b\in G$, if $a$ and $b$ satisfy

$$a^n = b^n \neq e$$

does it imply that $a=b$?

Answer 1

Having the equation $a^n=b^n$ hold for a single integer $n$ does not imply $a=b$. However, if it holds for two integers $n,m$ such that $\gcd(n,m)=1$, then it does follow that $a=b$.

Proof: By Bezout's lemma, there exists integers $x,y$ for which $mx+ny=1$. Then $$ a =a^{mx+ny}=(a^m)^x\cdot(a^n)^y=(b^m)^x\cdot(b^n)^y=b^{mx+ny}=b $$

Answer 2

Consider the Quaternions, then $i^2=j^2=-1$ but they are distinct elements

Answer 3

No, consider $\{1,i,-i,-1\}$, $i^2=(-i)^2=-1$