1.22: Conjugacy class
If g 1 and g 2 are two elements of a group G, they are called conjugate if there exists an element g 3 in G such that:
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- g 3 g 1 g 3 -1 = g 2 .
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Conjugacy is an equivalence relation and therefore partitions G into equivalence classes: every element of the group belongs to precisely one conjugacy class
The equivalence class that contains the element g 1 in G is
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- Cl(g 1 ) = { g 3 g 1 g 3 -1 | g 3 ∈ G}
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and is called the conjugacy class of g 1 . The class number of G is the number of conjugacy classes.
The classes Cl(g 1 ) and Cl(g 2 ) are equal if and only if g 1 and g 2 are conjugate, and disjoint otherwise.
For Abelian groups the concept is trivial, since each element forms a class on its own.