In which we meet the Orbit-Counting Formula
- Definition of
for an element
of a group
acting on a set.
- Theorem 65 (Orbit-Counting Formula): Let
be a finite group acting on a finite set
. Then
. We defined a set
and counted its elements in two ways.
- Lemma 66: Let
be a group acting on a finite set
. Take
,
with
and
conjugate. Then
. This was a quick check: we showed that for
we have
if and only if
.
- Corollary 67: Let
be a finite group acting on a finite set
. Say
has
conjugacy classes, and pick a representative from each, say
, …,
. Then
. This was immediate from Theorem 65 and Lemma 66.
for an element 
of a group 
acting on a set.
. Then 
. We defined a set 
and counted its elements in two ways.
, 
with 
conjugate. Then 
. This was a quick check: we showed that for 
we have 
if and only if 
.
conjugacy classes, and pick a representative from each, say 
. Then 
. This was immediate from Theorem 65 and Lemma 66.Understanding today’s lecture
Are you happy about why the Orbit-Counting Formula in Corollary 67 follows from the version of the Orbit-Counting Formula in Theorem 65?
You could check that the answer we obtained in the first example (about colourings of the edges of an equilateral triangle) matches what you obtained directly for 
What are the conjugacy classes in 
Further reading
The Orbit-Counting Formula has many names. It is sometimes known as Burnside‘s Lemma, although was not first proved by Burnside. It is relevant in Representation Theory. Here’s a page with some more applications of the result.
Preparation for Lecture 16
Sheet 7 Q5 is excellent preparation for Lecture 16.
What is the group of rotational symmetries of a cube? Or of a tetrahedron?
Why not build a cube and bring it along to the lecture for reference?