The $z$-Classes of Isometries

Details The $z$-Classes of Isometries The $z$-Classes of Isometries

2009-06-02

arxiv.org/abs/0906.0563v1

Mathematics

Group Theory

Scientific paper

Let $G$ be a group. Two elements $x,y$ are said to be in the same \emph{z-class} if their centralizers are conjugate in $G$. The conjugacy classes give a partition of $G$. Further decomposition of the conjugacy classes into $z$-classes provides an important information about the internal structure of the group. Let $\V$ be a vector space of dimension $n$ over a field $\F$ of characteristic different from 2. Let $B$ be a non-degenerate symmetric, or skew-symmetric, bilinear form on $\V$. Let $I(\V, B)$ denote the group of isometries of $(\V, B)$. We parametrize the $z$-classes in $I(\V, B)$. We show that the number of $z$-classes in $I(\V, B)$ is finite when $\F$ is perfect and has the property that it has only finitely many field extensions of degree $\leq n$. Along the way we also determine the conjugacy classes in $I(\V, B)$.

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