Finite Coxeter groups and their subgroup lattices (Q1073902)

The main result of the paper states that any finite Coxeter group of rank \(\geq 3\) is uniquely determined by its subgroup lattice. Moreover, any isomorphism of the subgroup lattice is induced by a group automorphism. The proof is based on the observation that the 2-element subgroups can be recognized in the subgroup lattice of these groups. Also, the case of rank 2 Coxeter groups, i.e. dihedral groups, is considered. If G is dihedral of order 2k and \(k=p_ 1^{e_ 1}...p_ r^{e_ r}\) the prime factorization with \(p_ 1<p_ 2<...<p_ r\), then G is uniquely determined by its subgroup lattice except in the following cases: (a) \(GCD(p_ 1-1,...,p_ r-1)\) has an odd prime divisor; or (b) k is odd, \(e_ 1=1\) and \(p_ 1\) divides \(GCD(p_ 2-1,...,p_ r-1)\). \{Remark. The result is incorrectly stated in the paper.\}