A characterization of subgroups of the orthogonal group
Let \(K\) be a field of characteristic distinct from 2. Put \(K^n=V\), assume \(n\geq 2\), and let \(G\) be a subgroup of the general linear group \(\text{GL}_n(K)\). Suppose \(G\) is generated by the set \(S\) of all involutions \(\sigma\in G\) satisfying \(\dim V(\sigma-1)=1\). Such an element \(\sigma\) is called a reflection. The group \(G\) is a reflection group if it has the following two properties: (a) \(-1_V\in G\), (b) an element \(\pi\) in \(G\) is a product of \(\dim V(\pi-1)\) elements \(\sigma\) in \(S\) whenever \(V(\pi-1)\) is not contained in the kernel \(\text{ker}(\pi-1)=F(\pi)\) of \(\pi-1\). It turns out that a reflection group is a subgroup of an orthogonal group if it contains a simplex. The authors consider also decompositions into irreducible reflection groups. A reflection group is reducible if \(S=S'\cup S''\) where \(S'\cap S''=\emptyset\); \(S',S''\neq\emptyset\), and \(\sigma'\sigma''=\sigma''\sigma'\) for all \(\sigma'\in S'\) and \(\sigma''\in S''\). For an irreducible reflection group \((G,S)\) put \(G'=\langle S'\rangle\), and \(G''=\langle S''\rangle\). Then \((G',S')\) and \((G'',S'')\) are reflection groups with \(G=G'\times G''\). Moreover, a reflection group \((G,S)\) which is the direct product of irreducible reflection groups such that each factor has dimension at most four, is a subgroup of an orthogonal group.
scientific article; zbMATH DE number 913033