The

centralizer of an element

*a* in a

group *G* consists
of all those elements of the group that

commute with

*a*
*C(a)={x* in

*G* such that

*xa=ax*}.

If the group is finite then we have the formula *|C(a)|.|Conj(a)|=|G|*,
where *Conj(a)* is the conjugacy class of *a*.

There is a similar notion for rings.