Constructing representations of finite groups and applications to finitely presented groups (Q1265516)

The group \(G^{3,7,n}\) is defined by the presentation \[ \langle x,y,z\mid x^2, y^2, z^2, (xy)^2, (xz)^3, (yz)^7, (xyz)^n\rangle. \] It is a famous problem whether the group \(G^{3,7,23}\) is infinite or not. Here is a very elegant proof that \(G^{3,7,23}\) is infinite by exploiting the representation theory of a finite factor group. The method is described in detail and applicable in a more general context.