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Concept
Conjugacy class
Summary
In mathematics, especially group theory, two elements and of a group are conjugate if there is an element in the group such that This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under for all elements in the group.
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 (singleton set).
Functions that are constant for members of the same conjugacy class are called class functions.
Let be a group. Two elements are conjugate if there exists an element such that in which case is called of and is called a conjugate of
In the case of the general linear group of invertible matrices, the conjugacy relation is called matrix similarity.
It can be easily shown that conjugacy is an equivalence relation and therefore partitions into equivalence classes. (This means that every element of the group belongs to precisely one conjugacy class, and the classes and are equal if and only if and are conjugate, and disjoint otherwise.) The equivalence class that contains the element is
and is called the conjugacy class of The of is the number of distinct (nonequivalent) conjugacy classes. All elements belonging to the same conjugacy class have the same order.
Conjugacy classes may be referred to by describing them, or more briefly by abbreviations such as "6A", meaning "a certain conjugacy class with elements of order 6", and "6B" would be a different conjugacy class with elements of order 6; the conjugacy class 1A is the conjugacy class of the identity which has order 1. In some cases, conjugacy classes can be described in a uniform way; for example, in the symmetric group they can be described by cycle type.
The symmetric group consisting of the 6 permutations of three elements, has three conjugacy classes:
No change .
Official source
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