In group theory, a discipline within modern algebra, an element of a group is called a real element of if it belongs to the same conjugacy class as its inverse , that is, if there is a in with , where is defined as .[1] An element of a group is called strongly real if there is an involution with .[2]

An element of a group is real if and only if for all representations of , the trace of the corresponding matrix is a real number. In other words, an element of a group is real if and only if is a real number for all characters of .[3]

A group with every element real is called an ambivalent group. Every ambivalent group has a real character table. The symmetric group of any degree is ambivalent.

Properties

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

For a real element   of a group  , the number of group elements   with   is equal to  ,[1] where   is the centralizer of  ,

 .

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   is real in   and   is odd, then   is strongly real in  .

Extended centralizer

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The extended centralizer of an element   of a group   is defined as

 

making the extended centralizer of an element   equal to the normalizer of the set  .[4]

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

 

See also

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Notes

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  1. ^ a b c Rose (2012), p. 111.
  2. ^ Rose (2012), p. 112.
  3. ^ a b Isaacs (1994), p. 31.
  4. ^ Rose (2012), p. 86.

References

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  • 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.