Some \(Y\)-groups (Q678573)

A \(Y\)-group is a group which is a factorgoup of a Coxeter group whose diagram has the shape of a \(Y\) with arms of length \(p,q,r\in\mathbb{N}\), also denoted by a \(Y_{pqr}\)-group. The most famous one might be the \(Y_{555}\)-group with the one additional relation \((ab_1c_1ab_2c_2ab_3c_3)^{10}=1\) which has been proved to be the bimonster by \textit{S. Norton} [Lond. Math. Soc. Lect. Note Ser. 165, 63-76 (1992; Zbl 0806.20019)] and \textit{A. Ivanov} [J. Algebra 163, No. 1, 88-108 (1994; Zbl 0797.20018)]. The author investigates the cases \(1=r\), \(1\leq p\leq 5\), \(q\leq 4\) and \(2=r\leq q\leq p\leq 5\), \(q\leq 3\) except \(pqr=532\). These groups are all related to some groups generated 3-transpositions. This basically is the idea of the proof. The author shows that the \(Y\)-presentation yields a so called Fischerian presentation which then can be used to identify the groups. Due to this fact no computer application is needed.