I am trying to determine whether $$\langle x, y\mid xyxy=1\rangle$$ is homomorphic to a known group. I cannot decide on a possible canditate from the groups I know and I cant seem to find this group on any list. Could someone help me me out?
2026-03-26 03:11:37.1774494697
Is this a presentation of a known group?
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Let $z = xy$. Note that $$\langle x, y : xyxy = 1\rangle = \langle x, z : z^2 = 1 \rangle$$ so your group is isomorphic to the free product $$ \mathbf Z \ast \mathbf Z/ 2\mathbf Z.$$