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Real element

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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 x g = x − 1 , where x g is defined as g − 1 ⋅ x ⋅ g . 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 .

Contents

An element x of a group G is called strongly real if there is an involution t with x t = x − 1 .

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.

kof tm the real element war


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 x g = x − 1 is equal to | C G ( x ) | , where C G ( x ) is the centralizer of x ,

C G ( x ) = { g ∈ G ∣ 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 and x 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 ∈ G ∣ x g = x ∨ 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, centralizer and extended centralizer are equal. For a real element x of a group G that is not an involution,

| C G ∗ ( x ) : C G ( x ) | = 2.

References

Real element Wikipedia
(Text) CC BY-SA


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