Real element

In group theory, a discipline within modern algebra, an element formula_1 of a group formula_2 is called a real element of formula_2 if it belongs to the same conjugacy class as its inverse formula_4, that is, if there is a formula_5 in formula_2 with formula_7, where formula_8 is defined as formula_9. An element formula_1 of a group formula_2 is called strongly real if there is an involution formula_12 with formula_13.
An element formula_1 of a group formula_2 is real if and only if for all representations formula_16 of formula_2, the trace formula_18 of the corresponding matrix is a real number. In other words, an element formula_1 of a group formula_2 is real if and only if formula_21 is a real number for all characters formula_22 of formula_2.
A group with every element real is called an ambivalent group. Every ambivalent group has a real character table. The symmetric group formula_24 of any degree formula_25 is ambivalent.
Properties.
A group with real elements other than the identity element necessarily is of even order.
For a real element formula_1 of a group formula_2, the number of group elements formula_5 with formula_7 is equal to formula_30, where formula_31 is the centralizer of formula_1,
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 formula_1 is real in formula_2 and formula_30 is odd, then formula_1 is strongly real in formula_2.
Extended centralizer.
The extended centralizer of an element formula_1 of a group formula_2 is defined as
making the extended centralizer of an element formula_1 equal to the normalizer of the set 
The extended centralizer of an element of a group formula_2 is always a subgroup of formula_2. For involutions or non-real elements, centralizer and extended centralizer are equal. For a real element formula_1 of a group formula_2 that is not an involution,