# What is this group $G=\langle a,b,c\mid a^2=1, b^2=1, c^2=ab\rangle$

Consider the group presentation $$G=\langle a,b,c\mid a^2=1, b^2=1, c^2=ab\rangle.$$

Is this a known group? What is $G$ isomorphic to?

Thanks a lot.

• It reminds me of a triangle group. – Shaun May 24 at 13:56
• One can eliminate $b$: $$G\cong\langle a,c\mid a^2, (c^2a^{-1})^2\rangle.$$ Presentations whose relators are proper powers of some words are widely studied . . . – Shaun May 24 at 14:03
• Sorry: $$G\cong\langle a,c\mid a^2, (a^{-1}c^2)^2\rangle.$$ – Shaun May 25 at 12:22

For example it is virtually abelian: the subgroup $H = \langle ab, cac^{-1}a \rangle$ is free abelian and normal in $G$ with $G/H \cong C_2^2$.
You can see directly from the presentation that the subgroup $\langle ab \rangle$ is normal with quotient group the infinite dihedral group generated by the images of $a$ and $c$.
Based on $$G\cong\langle a,c\mid a^2, (a^{-1}c^2)^2\rangle,$$ we have
\begin{align} G&\cong\langle a, c\mid a^2, ac^2a^{-1}=c^{-2}\rangle \\ &\cong \boxed{BS(2,-2)/\Bbb Z_2}, \end{align}
where $$BS(2,-2)$$ is a Baumslag–Solitar group and the normal subgroup $$\Bbb Z_2$$ is generated by $$a$$.