TITLE: Prove that $xy=yx$
QUESTION [8 upvotes]: Let $(G,\cdot)$ be a group and $f,g:G\rightarrow G,f(x)=x^6,g(x)=x^{10}$ are homomorphisms, where $f$ is injective. Prove that $xy=yx$.

We know that$(xy)^{10}=x^{10}y^{10},(xy)^{6}=x^{6}y^{6},(xy)^{9}=y^{9}x^{9},(xy)^{5}=y^{5}x^{5},x^{9}y^{10}=y^{10}x^{9}$ and $x^{5}y^{6}=y^{6}x^{5}$, but I couldn't use these results in a productive way.

REPLY [3 votes]: This statement is false. Take $UT(3, 5)$ as a counterexample. Any other non-abelian group of exponent $5$ will do too.