Kerodon

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Remark 2.4.7.2. We will use the term Coxeter group to refer to a group $W$ together with a choice of subset $S \subseteq W$ for which the pair $(W,S)$ is a Coxeter system. Beware that the subset $S$ is not determined by the structure of $W$ as an abstract group: for example, if $(W,S)$ is a Coxeter system, then so is $(W, wSw^{-1})$ for each $w \in W$. In other words, a Coxeter group is not merely a group, but a group equipped with some additional structure (namely, the structure of a Coxeter system $(W,S)$).