Real element

In group theory, a discipline within modern algebra, an element [math]\displaystyle{ x }[/math] of a group [math]\displaystyle{ G }[/math] is called a real element of [math]\displaystyle{ G }[/math] if it belongs to the same conjugacy class as its inverse [math]\displaystyle{ x^{-1} }[/math], that is, if there is a [math]\displaystyle{ g }[/math] in [math]\displaystyle{ G }[/math] with [math]\displaystyle{ x^g = x^{-1} }[/math], where [math]\displaystyle{ x^g }[/math] is defined as [math]\displaystyle{ g^{-1} \cdot x \cdot g }[/math]. An element [math]\displaystyle{ x }[/math] of a group [math]\displaystyle{ G }[/math] is called strongly real if there is an involution [math]\displaystyle{ t }[/math] with [math]\displaystyle{ x^t = x^{-1} }[/math]. An element [math]\displaystyle{ x }[/math] of a group [math]\displaystyle{ G }[/math] is real if and only if for all representations [math]\displaystyle{ \rho }[/math] of [math]\displaystyle{ G }[/math], the trace [math]\displaystyle{ \mathrm{Tr}(\rho(g)) }[/math] of the corresponding matrix is a real number. In other words, an element [math]\displaystyle{ x }[/math] of a group [math]\displaystyle{ G }[/math] is real if and only if [math]\displaystyle{ \chi(x) }[/math] is a real number for all characters [math]\displaystyle{ \chi }[/math] of [math]\displaystyle{ G }[/math].

A group with every element real is called an ambivalent group. Every ambivalent group has a real character table. The symmetric group [math]\displaystyle{ S_n }[/math] of any degree [math]\displaystyle{ n }[/math] is ambivalent.

Properties

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

For a real element [math]\displaystyle{ x }[/math] of a group [math]\displaystyle{ G }[/math], the number of group elements [math]\displaystyle{ g }[/math] with [math]\displaystyle{ x^g = x^{-1} }[/math] is equal to [math]\displaystyle{ \left|C_G(x)\right| }[/math], where [math]\displaystyle{ C_G(x) }[/math] is the centralizer of [math]\displaystyle{ x }[/math],

[math]\displaystyle{ \mathrm{C}_G(x) = \{ g \in G\mid x^g = x \} }[/math].

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 [math]\displaystyle{ x \ne e }[/math] and [math]\displaystyle{ x }[/math] is real in [math]\displaystyle{ G }[/math] and [math]\displaystyle{ \left|C_G(x)\right| }[/math] is odd, then [math]\displaystyle{ x }[/math] is strongly real in [math]\displaystyle{ G }[/math].

Extended centralizer

The extended centralizer of an element [math]\displaystyle{ x }[/math] of a group [math]\displaystyle{ G }[/math] is defined as

[math]\displaystyle{ \mathrm{C}^*_G(x) = \{ g \in G\mid x^g = x \lor x^g = x^{-1} \}, }[/math]

making the extended centralizer of an element [math]\displaystyle{ x }[/math] equal to the normalizer of the set [math]\displaystyle{ \left\{x, x^{-1}\right\} }[/math].

The extended centralizer of an element of a group [math]\displaystyle{ G }[/math] is always a subgroup of [math]\displaystyle{ G }[/math]. For involutions or non-real elements, centralizer and extended centralizer are equal. For a real element [math]\displaystyle{ x }[/math] of a group [math]\displaystyle{ G }[/math] that is not an involution,

[math]\displaystyle{ \left|\mathrm{C}^*_G(x):\mathrm{C}_G(x)\right| = 2. }[/math]

See also

• Brauer–Fowler theorem

References

• Gorenstein, Daniel (2007). Finite Groups. AMS Chelsea Publishing. ISBN 978-0821843420. 
• Isaacs, I. Martin (1994). Character Theory of Finite Groups. Dover Publications. ISBN 978-0486680149. 
• Rose, John S. (2012). A Course on Group Theory. Dover Publications. ISBN 978-0-486-68194-8.