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International Journal of Mathematics and Mathematical Sciences
Volume 2003, Issue 20, Pages 1241-1249

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 Hindawi Publishing Corporation. 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.


A subgroup H of a group G is called malnormal in G if it satisfies the condition that if gG and hH, h1 such that ghg1H, then gH. 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.