Real element explained
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
.
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
=\{g\inG\midxg=x\lorxg=x-1\},
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,
See also
References
- Book: Gorenstein . Daniel . Daniel Gorenstein . Finite Groups . AMS Chelsea Publishing . 978-0821843420 . 2007 . reprint of a work originally published in 1980.
- Book: Isaacs, I. Martin . Martin Isaacs . Character Theory of Finite Groups . Dover Publications . 978-0486680149 . 1994 . unabridged, corrected republication of the work first published by Academic Press, New York in 1976 .
- Book: Rose, John S. . 2012 . A Course on Group Theory . Dover Publications . 978-0-486-68194-8 . unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978 .