In group theory, a discipline within modern algebra, an element
is called a real element of
if it belongs to the same conjugacy class as its inverse
of a group
is called strongly real if there is an involution
of a group
is real if and only if for all representations
, the trace
{\displaystyle \mathrm {Tr} (\rho (g))}
of the corresponding matrix is a real number.
In other words, an element
of a group
is a real number for all characters
[3] 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
A group with real elements other than the identity element necessarily is of even order.
[3] For a real element
, the number of group elements
, 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.
is strongly real in
The extended centralizer of an element
is defined as making the extended centralizer of an element
equal to the normalizer of the set
[4] The extended centralizer of an element of a group
For involutions or non-real elements, centralizer and extended centralizer are equal.
[1] For a real element