Real element explained

In group theory, a discipline within modern algebra, an element

x

of a group

G

is called a real element of

G

if it belongs to the same conjugacy class as its inverse

x-1

, that is, if there is a

g

in

G

with

xg=x-1

, where

xg

is defined as

g-1xg

. An element

x

of a group

G

is called strongly real if there is an involution

t

with

xt=x-1

.

An element

x

of a group

G

is real if and only if for all representations

\rho

of

G

, the trace

Tr(\rho(g))

of the corresponding matrix is a real number. In other words, an element

x

of a group

G

is real if and only if

\chi(x)

is a real number for all characters

\chi

of

G

.

Sn

of any degree

n

is ambivalent.

Properties

A group with real elements other than the identity element necessarily is of even order.

For a real element

x

of a group

G

, the number of group elements

g

with

xg=x-1

is equal to

\left|CG(x)\right|

, where

CG(x)

is the centralizer of

x

,

CG(x)=\{g\inG\midxg=x\}

.

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

x

is real in

G

and

\left|CG(x)\right|

is odd, then

x

is strongly real in

G

.

Extended centralizer

The extended centralizer of an element

x

of a group

G

is defined as
*
C
G(x)

=\{g\inG\midxg=x\lorxg=x-1\},

making the extended centralizer of an element

x

equal to the normalizer of the set

The extended centralizer of an element of a group

G

is always a subgroup of

G

. For involutions or non-real elements, centralizer and extended centralizer are equal. For a real element

x

of a group

G

that is not an involution,
*
\left|C
G(x):C

G(x)\right|=2.

See also

References