Real element

In mathematics, specifically group theory, 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 $g^{- 1} x g = x^{- 1}$ . The element $x$ is called strongly real if there is an involution $t$ with $x^{t} = x^{- 1}$ .

An element $x$ of a group $G$ is real if and only if for all representations $ρ$ of $G$ , the trace $T r (ρ (g))$ of the corresponding matrix is a real number. In other words, an element $x$ of a group $G$ is real if and only if $χ (x)$ is a real number for all characters $χ$ of $G$ .

A group with every element real is called an ambivalent group. Every ambivalent group has a real character table. The symmetric group $S_{n}$ 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 $g^{- 1} x g = x^{- 1}$ is equal to $| C_{G} (x) |$ , where $C_{G} (x)$ is the centralizer of $x$ :

C_{G} (x) = {g \in G ∣ g^{- 1} 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 $x \neq e$ is real in $G$ and $| C_{G} (x) |$ 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 \in G ∣ g^{- 1} x g = x \lor g^{- 1} x g = x^{- 1}}

making the extended centralizer of an element $x$ equal to the normalizer of the set ${x, x^{- 1}}$ .

The extended centralizer of an element of a group $G$ is always a subgroup of $G$ . For involutions or non-real elements, the centralizer and extended centralizer are equal. For a real element $x$ that is not an involution,

| C_{G}^{*} (x) : C_{G} (x) | = 2.

References

Citations

1 2 3 Rose (1994), p. 111.
↑ Rose (1994), p. 112.
1 2 Isaacs (1994), p. 31.
↑ Rose (1994), p. 86.

Sources

Gorenstein, Daniel (2007) [1968]. Finite Groups (reprint of 1980 2nd ed.). Providence, RI: AMS Chelsea. ISBN 978-0-8218-4342-0.

Isaacs, I. Martin (1994) [1976]. Character Theory of Finite Groups (reprint ed.). Mineola, NY: Dover. ISBN 978-0-486-68014-9.

Rose, John S. (1994) [1978]. A Course on Group Theory (reprint ed.). Mineola, NY: Dover. ISBN 978-0-486-68194-8. Zbl 848.20001.

[FOOTNOTERose1994111-1] 1 2 3 Rose (1994), p. 111.

[FOOTNOTERose1994112-2] Rose (1994), p. 112.

[FOOTNOTEIsaacs199431-3] 1 2 Isaacs (1994), p. 31.

[FOOTNOTERose199486-4] Rose (1994), p. 86.