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In mathematics, especially group theory, two elements a {\displaystyle a} and b {\displaystyle b} of a group are conjugate if there is an element g {\displaystyle g} in the group such that b = g − 1 a g . {\displaystyle b=g^{-1}ag.} This is an equivalence relation whose equivalence classes are called conjugacy classes.
Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element.
Functions that are constant for members of the same conjugacy class are called class functions.