Conjugation

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Definition

Let G be a group and g \in G be an element. Then, the conjugation map by g, denoted c_g, is defined as the map:

x \mapsto gxg^{-1}.

In other words, c_g(x) = gxg^{-1}.

Note that when the convention is to make the group act on the right, conjugation by g is defined as:

x \mapsto g^{-1}xg

and further, this is denoted as x^g.

Facts

Related terms

  • Inner automorphism: An automorphism that can be expressed as c_g for some g \in G.
  • Conjugate elements: Two elements x,y \in G are termed conjugate if there exists g \in G such that gxg^{-1} = y.
  • Conjugacy class: The conjugacy class of x \in G is the set of all elements that can be written as gxg^{-1} for some g \in G.