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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.

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  • $\begingroup$ It reminds me of a triangle group. $\endgroup$ – Shaun May 24 at 13:56
  • $\begingroup$ 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 . . . $\endgroup$ – Shaun May 24 at 14:03
  • $\begingroup$ Sorry: $$G\cong\langle a,c\mid a^2, (a^{-1}c^2)^2\rangle.$$ $\endgroup$ – Shaun May 25 at 12:22
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I don't know about "known groups". I can answer questions about its properties.

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 $c$.

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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 $a$.

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