International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 20, Pages 1241-1249
doi:10.1155/S0161171203205305
On centralizers of elements of groups acting on trees with
inversions
College of Education and Basic Science, Ajman University of Science and Technology, Abu Dhabi, United Arab Emirates
Received 2 May 2002
Copyright © 2003 R. M. S. Mahmood. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A subgroup H of a group G is called malnormal in G if it satisfies the condition that if g∈G and h∈H, h≠1
such that ghg−1∈H, then g∈H. In this paper, we show
that if G is a group acting on a tree X with inversions such
that each edge stabilizer is malnormal in G, then the
centralizer C(g) of each nontrivial element g of G is in a
vertex stabilizer if g is in that vertex stabilizer. If g is
not in any vertex stabilizer, then C(g) is an infinite cyclic
if g does not transfer an edge of X to its inverse. Otherwise,
C(g) is a finite cyclic of order 2.