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.