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The size of a conjugacy class equals the index of the centralizer
Conjugacy Class Size Lemma: Let be a finite group and let . Let denote the conjugacy class of , and let denote the centralizer of . Then
where is the index of in .
This is an instance of the orbit–stabilizer theorem applied to the conjugation action of on itself.
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 subgroup of a group . The index of in , denoted , is the cardinality of the set of left cosets of in :
(Equivalently, it is the number of distinct right cosets.)
For finite , Lagrange's theorem implies , so in particular divides . Small index has strong consequences: iff , and if then is normal, i.e. a normal subgroup.
Orbit–Stabilizer Theorem. Let be a group acting on a set via a group action. For , let be the orbit of and let be the stabilizer of . Then the map
is a bijection. In particular, if is finite then
Let be a group. The conjugation action of on itself is the group action defined by