Real element


In group theory, a discipline within modern algebra, an element of a group is called a real element of if it belongs to the same conjugacy class as its inverse, that is, if there is a in with, where is defined as. An element of a group is called strongly real if there is an involution with.
An element of a group is real if and only if for all representations of, the trace of the corresponding matrix is a real number. In other words, an element of a group is real if and only if is a real number for all characters of.
A group with every element real is called an ambivalent group. Every ambivalent group has a real character table. The symmetric group of any degree is ambivalent.

Properties

A group with real elements other than the identity element necessarily is of even order.
For a real element of a group, the number of group elements with is equal to, where is the centralizer of,
Every involution is strongly real. Furthermore, every element that is the product of two involutions is strongly real. Conversely, every strongly real element is the product of two involutions.
If and is real in and is odd, then is strongly real in.

Extended centralizer

The extended centralizer of an element of a group is defined as
making the extended centralizer of an element equal to the normalizer of the set
The extended centralizer of an element of a group is always a subgroup of. For involutions or non-real elements, centralizer and extended centralizer are equal. For a real element of a group that is not an involution,