TITLE: Is this group abelian?
QUESTION [8 upvotes]: I tried to show that the following group is abelian by manipulation the relations but they didn't work. Please show me the right way. The group is $$G:=\left<x,y \mid xyxy^2=yxyx^2=1\right>$$

REPLY [7 votes]: Hint: You can identify $xyxy^2$ as a subword of $yxyx^2$. In details:

 $$\begin{array}{ll} yxyx^2=1 & \Rightarrow xyxyx^2=x \\ & \Rightarrow (xyxy^2)y^{-1}x^2=x \\ & \Rightarrow y^{-1}x^2=x \\ & \Rightarrow y^{-1}x=1 \\ & \Rightarrow x=y \end{array}$$

So $G \simeq \langle x \mid x^5=1 \rangle \simeq \mathbb{Z}_5$.