An example of generalized Coxeter group which fails to be the even subgroup of a Coxeter group (Q1204624)

A group \(G\) is called a generalized Coxeter group if it has a presentation of the form: \(G = \langle x_ 1,\dots,x_ n;\;x^{q_ k}_ k,(x^{\alpha_{ij}}_ ix^{\beta_{ij}}_ j)^{q_{ij}}\), \((\alpha_{ij},q_ i) = (\beta_{ij},q_ j) = 1\), \(1\leq i < j \leq n\rangle\). In the work of \textit{S. Tsaranov} [Algebras Groups Geom. 6, No. 3, 281-318 (1989; Zbl 0702.20020)], the following question was raised: is it true that every generalized Coxeter group is a Coxeter group or an even subgroup of a suitable Coxeter group? The present article gives a negative answer to this question.