Conjugacy classes of centralizers in unitary groups

Sushil Bhunia, Anupam Singh

Published 2016-10-21Version 1

Let G be a group. Two elements x and y in G are said to be in the same z-class if their centralisers in G are conjugate within G. Consider F a perfect field of characteristic not 2 which has a non-trivial Galois automorphism of order 2. Further, suppose that the fixed field has the property that it has only finitely many field extensions of any finite degree. In this paper, we prove that the number of z-classes in unitary group over such fields is finite. Further, we count the number of z-classes in the finite unitary group and prove that this number is same as that of general linear group.