Three-dimensional classical groups acting on polytopes (Q603843)

The authors show that any irreducible subgroup \(G\) of \(\mathrm{GL}(V)\), where \(V\) is a three-dimensional vector space over a finite field \(K\), that arises as automorphism group of an abstract regular polytope preserves a non-degenerate symmetric bilinear form \(f\) of \(V\). In any such case \(K\) is of characteristic \(\neq 2\). Moreover, let \(I(V,f)\leq\mathrm{GL}(V)\) be the group of \(f\)-isometries of \(V\). Then \(I(V,f)'\leq G\leq I(V,f)\), where \(I(V,f)'\) is the commutator subgroup of \(I(V,f)\).