Difference between revisions of "Conjugacy class"
From Online Dictionary of Crystallography
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| + | <font color="blue">Classe de conjugaison</font> (''Fr''); <font color="red">Konjugationsklasse</font> (''Ge''); <font color="black">Classe coniugata</font> (''It''); | ||
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If g<sub>1</sub> and g<sub>2</sub> are two elements of a group G, they are called '''conjugate''' if there exists an element g<sub>3</sub> in G such that: | If g<sub>1</sub> and g<sub>2</sub> are two elements of a group G, they are called '''conjugate''' if there exists an element g<sub>3</sub> in G such that: | ||
Revision as of 11:47, 27 February 2007
Classe de conjugaison (Fr); Konjugationsklasse (Ge); Classe coniugata (It);
If g1 and g2 are two elements of a group G, they are called conjugate if there exists an element g3 in G such that:
- g3g1g3-1 = g2.
Conjugacy is an equivalence relation and therefore partitions G into equivalence classes: every element of the group belongs to precisely one conjugacy class; the classes Cl(g1) and Cl(g2) are equal if and only if g1 and g2 are conjugate, and disjoint otherwise.
The equivalence class that contains the element g1 in G is
- Cl(g1) = { g3g1g3-1| g3 ∈ G}
and is called the conjugacy class of g1. The class number of G is the number of conjugacy classes.
For abelian groups the concept is trivial, since each class is a set of one element.