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Given two elements a and b such that $a = a^{-1}$ and $aba = b^{-1}$, what's the order of the subgroup generated by $<a, b>$? I'm having trouble coming up with any relevant theorems that could help me with this.

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You can't tell from just that information.

In the one extreme, if there are no further relations, i.e. you're looking at the quotient of the free group on $\{a,b\}$ by the normal subgroup generated by those relations, this is an infinite group with elements of the form $b^n$, $ab^n$ and $b^na$. In the other extreme, if $b=b^{-1}$, this is the Klein four-group.

Note that $ab=b^{-1}a^{-1}=(ab)^{-1}$ and that $b=a(ab)$, so the group is also generated by the two elements $a$ and $ab$ of order two, so that all elements can be written as alternating products of $a$ and $ab$ and all products $a(ab)$ can be replaced by $b$, leading to the above characterization.

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