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conjugacy class (Definition)

Two elements $ g$ and $ g'$ of a group $ G$ are said to be conjugate if there exists $ h \in G$ such that $ g' = hgh^{-1}$. Conjugacy of elements is an equivalence relation, and the equivalence classes of $ G$ are called conjugacy classes.

Two subsets $ S$ and $ T$ of $ G$ are said to be conjugate if there exists $ g \in G$ such that

$\displaystyle T = \{gsg^{-1} \mid s \in S\} \subset G. $
In this situation, it is common to write $ gSg^{-1}$ for $ T$ to denote the fact that everything in $ T$ has the form $ gsg^{-1}$ for some $ s \in S$. We say that two subgroups of $ G$ are conjugate if they are conjugate as subsets.



"conjugacy class" is owned by djao.
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See Also: class equation

Other names:  conjugate, conjugate set, conjugate subgroup
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Cross-references: subgroups, subsets, equivalence classes, equivalence relation, group
There are 42 references to this entry.

This is version 2 of conjugacy class, born on 2002-02-07, modified 2002-07-25.
Object id is 1848, canonical name is ConjugacyClass.
Accessed 13772 times total.

Classification:
AMS MSC: 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
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