Order of the Mennicke group M(a,b,c)

Determine the exact order of the three-generator, three-relation Mennicke group M(a,b,c) for general integer parameters a, b, c, beyond the currently known upper bounds and special cases where finiteness or specific orders are established.

Background

The paper reviews prior work on balanced presentations, highlighting Mennicke’s family M(a,b,c) of finite groups with three generators and three relations. While several authors have established finiteness criteria and provided upper bounds for the order of M(a,b,c), a complete determination of the order across general parameters has remained elusive.

Within this context, the authors explicitly note that despite substantial examination in the literature, the exact order of M(a,b,c) is not generally known, referencing work by Mennicke, Johnson–Robertson, Albar–Al-Shuaibi, and Jabara.