Suppose $G$ has the presentation $\langle t, x\_1, x\_2, ... | R \rangle$ where each relator in $R$ has the form $t^{-1}x\_it = x\_j$ for some $i,j$. Does $G$ have an element of order 2?


This is an HNN extension of a free group, if that changes anything.



Does the group given by this presentation have an element of order 2?
By the torsion theorem for HNN-extensions, every element of finite order is conjugate to an element of the base, which in your case is a free group, so the answer is no.


See Lyndon and Schupp's Combinatorial Group Theory.


