The groups in question are generated by elements A and B subject to the relations

in which α and β are fixed integers. We prove:

Theorem. Each group of the class just presented is finite when neither α nor β equals 1, and is nilpotent. Its order is a factor of 27 (α — 1) (β — l)∈8where ∈ is the greatest common divisor of α — 1 and β — 1, and its nilpotency class is at most 8.

We denote the commutator X-1Y-1XY by [X, Y], and Y-1XY by XY. If X1, X2, … , Xr are elements of some group then {X1, X2, … , Xr} will mean the subgroup which they generate.