On a Class of Finitely Presented Groups

Extract

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 X^Y. If X₁, X₂, … , X_r are elements of some group then {X₁, X₂, … , X_r} will mean the subgroup which they generate.