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A group acts on itself by conjugation g·x = gxg^{-1}
Proposition (Conjugation action). Let be a group. Define a map by
Then this defines a group action of on itself, called the conjugation action.
Context. The orbits of this action are the conjugacy classes in , and stabilizers are centralizers. This action is the mechanism behind the class equation and many counting arguments.
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
Let be a group. The conjugation action of on itself is the group action
Let a group action of a group on a set be given, written . For , the orbit of is the subset
Let a group action of a group on a set be given. For , the stabilizer of is
The stabilizer is always a subgroup of .
Let be a finite group. For , set
and let the centralizer be
Then . If represent the noncentral conjugacy classes, the class equation is