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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}. 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}}.
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 ] {\displaystyle \mathrm {Tr} ]} 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 {\displaystyle \chi } is a real number for all characters χ {\displaystyle \chi } of G {\displaystyle 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 {\displaystyle S_{n}} of any degree n {\displaystyle n} is ambivalent.