Class equation of a group
From Groupprops
Contents
Statement
Suppose is a finite group,
is the center of
, and
are all the conjugacy classes in
comprising the elements outside the center. Let
be an element in
for each
. Then, we have:
.
Note that this is a special case of the class equation of a group action where the group acts on itself by conjugation.
Related facts
Facts used
Proof
The proof follows directly from fact (1), and the following observations:
- When a group acts on itself by conjugation, the set of fixed points under the action is precisely the center of the group.
- The stabilizer of a point
under the action by conjugation is precisely the centralizer of
.