Almost central involutions in split extensions of Coxeter groups by graph automorphisms
نویسنده
چکیده
In this paper, given a split extension of an arbitrary Coxeter group by automorphisms of the Coxeter graph, we determine the involutions in that extension whose centralizer has finite index. Our result has applications to many problems such as the isomorphism problem of general Coxeter groups. In the argument, some properties of certain special elements and of the fixed-point subgroups by graph automorphisms in Coxeter groups, which are of independent interest, are also given.
منابع مشابه
Automorphisms of nearly finite Coxeter groups
Suppose that W is an infinite Coxeter group of finite rank n, and suppose that W has a finite parabolic subgroup WJ of rank n 1. Suppose also that the Coxeter diagram of W has no edges with infinite labels. Then any automorphism of W that preserves reflections lies in the subgroup of AutðWÞ generated by the inner automorphisms and the automorphisms induced by symmetries of the Coxeter graph. If...
متن کاملAutomorphisms of Coxeter Groups of Rank Three
If W is an infinite rank 3 Coxeter group, whose Coxeter diagram has no infinite bonds, then the automorphism group of W is generated by the inner automorphisms and any automorphisms induced from automorphisms of the Coxeter diagram. Indeed Aut(W ) is the semi-direct product of Inn(W ) and the group of graph automorphisms.
متن کاملFixed Points of Involutive Automorphisms of the Bruhat Order
Applying a classical theorem of Smith, we show that the poset property of being Gorenstein∗ over Z2 is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every interval in the Bruhat order on (twisted) involutions in an arbitrary Coxeter group has this property, and we find the rank function. This implies results conjectured by F. ...
متن کاملAutomorphisms of Right-Angled Coxeter Groups
If (W, S) is a right-angled Coxeter group, then Aut(W ) is a semidirect product of the group Aut◦(W ) of symmetric automorphisms by the automorphism group of a certain groupoid. We show that, under mild conditions, Aut◦(W ) is a semidirect product of Inn(W ) by the quotient Out◦(W ) = Aut◦(W )/Inn(W ). We also give sufficient conditions for the compatibility of the two semidirect products. When...
متن کاملOn Involutive Anti-Automorphisms of Finite Abelian Groups
We investigate various aspects of involutions of groups, i.e, anti-automorphisms of order at most two. The emphasis is on finite abelian groups. We count the number of involutions for the cyclic groups, and consider the problem for direct products of groups. We also give a characterization for the set of skewed squares of finitely generated abelian groups with identity as the involution. The pr...
متن کامل