Fundamental group of a 3-manifold (Q2640942)

Theorem: The group \(G(p)=<x,y|\) \(x^ 3=y^ 3\), \(x^{3p}=(zx^{- 1})^ 2>\) has order 72p for every \(p\geq 1\). These groups are the fundamental groups of certain 3-manifolds what makes them interesting. For odd p or \(p\leq 11\) the above result has been obtained by Lonergan- Hosack. The main tool for the proof is the Reidemeister-Schreier method for calculating the commutator subgroup. The factor group is cyclic of order 9p and the commutator subgroup is isomorphic to the group \(<v_ 0,v_ 1,v_ 2|\) \(v_ 0v_ 1=v_ 2\), \(v_ 1v_ 2=v_ 0\), \(v_ 2v_ 0=v_ 1\), \([v^ 2_ i,v_ j]=1\) \(0\leq i,j\leq 2>\) of order 8 (the multiplicative group generated by the quaternions of order 4).