Counting $(1,\beta )$-BM relations and classifying $(2,2)$-BM groups

Abstract

In the first part, we prove that the number of $(1,\beta )$-BM relations is $3\cdot 5\cdot \dots \cdot (2\beta +1)$, which was conjectured by Kimberley. In the second part, we construct two isomorphisms between certain $(2,2)$-BM groups. This completes the classification of $(2,2)$-BM groups initiated by Kimberley and Robertson.

11 pages