Real element

In group theory, a discipline within modern algebra, an element x {\displaystyle x} of a group G {\displaystyle G} is called a real element of G {\displaystyle G} if it belongs to the same conjugacy class as its inverse x − 1 {\displaystyle x^{-1}} , that is, if there is a g {\displaystyle g} in G {\displaystyle G} with x g = x − 1 {\displaystyle x^{g}=x^{-1}} , where x g {\displaystyle x^{g}} is defined as g − 1 ⋅ x ⋅ g {\displaystyle g^{-1}\cdot x\cdot g} .^[1] An element x {\displaystyle x} of a group G {\displaystyle G} is called strongly real if there is an involution t {\displaystyle t} with x t = x − 1 {\displaystyle x^{t}=x^{-1}} .^[2]

An element x {\displaystyle x} of a group G {\displaystyle G} is real if and only if for all representations ρ {\displaystyle \rho } of G {\displaystyle G} , the trace T r ( ρ ( g ) ) {\displaystyle \mathrm {Tr} (\rho (g))} of the corresponding matrix is a real number. In other words, an element x {\displaystyle x} of a group G {\displaystyle G} is real if and only if χ ( x ) {\displaystyle \chi (x)} is a real number for all characters χ {\displaystyle \chi } of G {\displaystyle G} .^[3]

A group with every element real is called an ambivalent group. Every ambivalent group has a real character table. The symmetric group S n {\displaystyle S_{n}} of any degree n {\displaystyle n} is ambivalent.

Properties

A group with real elements other than the identity element necessarily is of even order.^[3]

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

C G ( x ) = { g ∈ G ∣ x g = x } {\displaystyle \mathrm {C} _{G}(x)=\{g\in G\mid x^{g}=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 x ≠ e {\displaystyle x\neq e} and x {\displaystyle x} is real in G {\displaystyle G} and | C G ( x ) | {\displaystyle \left|C_{G}(x)\right|} is odd, then x {\displaystyle x} is strongly real in G {\displaystyle G} .

Extended centralizer

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

C G ∗ ( x ) = { g ∈ G ∣ x g = x ∨ x g = x − 1 } , {\displaystyle \mathrm {C} _{G}^{*}(x)=\{g\in G\mid x^{g}=x\lor x^{g}=x^{-1}\},}

making the extended centralizer of an element x {\displaystyle x} equal to the normalizer of the set { x , x − 1 } {\displaystyle \left\{x,x^{-1}\right\}} .^[4]

The extended centralizer of an element of a group G {\displaystyle G} is always a subgroup of G {\displaystyle G} . For involutions or non-real elements, centralizer and extended centralizer are equal.^[1] For a real element x {\displaystyle x} of a group G {\displaystyle G} that is not an involution,

| C G ∗ ( x ) : C G ( x ) | = 2. {\displaystyle \left|\mathrm {C} _{G}^{*}(x):\mathrm {C} _{G}(x)\right|=2.}

See also

• Brauer–Fowler theorem

Notes

References

• Gorenstein, Daniel (2007) [reprint of a work originally published in 1980]. Finite Groups. AMS Chelsea Publishing. ISBN 978-0821843420.
• Isaacs, I. Martin (1994) [unabridged, corrected republication of the work first published by Academic Press, New York in 1976]. Character Theory of Finite Groups. Dover Publications. ISBN 978-0486680149.
• Rose, John S. (2012) [unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978]. A Course on Group Theory. Dover Publications. ISBN 978-0-486-68194-8.