William Burnside
1. Group Actions and Orbits
So far a group has been a self-contained thing: a set with a multiplication table, plus whatever structure we could find inside it.But in post 1 the group came from somewhere (it was the set of symmetries of a triangle). Now we generalize, and let a group act on a set.
1.1 What an action is
A group \(G\) acts on a set \(X\) if every \(g \in G\) gives a way of moving the elements of \(X\) around, which follows the group operation. Specifically we need a rule \(g \cdot x\) satisfying
\[ e \cdot x = x \qquad \text{and} \qquad (gh) \cdot x = g \cdot (h \cdot x) \tag{1} \]
for all \(g, h \in G\) and \(x \in X\). Each \(g\) sends \(X\) to itself, and it's invertible because \(g^{-1}\) undoes it, so each \(g\) gives a permutation of \(X\). The second condition says that assigning permutations to group elements respects multiplication. So an action is nothing more than a homomorphism
\[ \varphi: G \to \operatorname{Sym}(X) \tag{2} \]
into the group of all permutations of \(X\).
1.2 The triangle again!
Let \(D_3\) act on its three corners, \(X = {1, 2, 3}\). From post 1, \(r\) sends \(1 \to 2 \to 3 \to 1\), and \(f\) holds corner 1 still while swapping 2 and 3.
Two questions you can ask about any point $x \in X
Where can it go? The set of everything reachable is the orbit:
\[ \operatorname{Orb}(x) = { g \cdot x : g \in G } \tag{3} \]
For corner 1, we have \(e \cdot 1 = 1\), \(r \cdot 1 = 2\), \(r^2 \cdot 1 = 3\), so the orbit is all of \({1, 2, 3}\). Every corner can be sent to every other corner, which makes sense (that's what it means for the triangle to be equilateral).
What holds it still? The set of group elements fixing \(x\) is the stabilizer:
\[ \operatorname{Stab}(x) = { g \in G : g \cdot x = x } \tag{4} \]
For corner 1: \(e\) fixes it, and \(f\) fixes it by construction, but the rotations move it. And from post 1, \(rf\) swaps corners 1 and 2, while \(r^2f\) swaps 1 and 3, so neither fixes corner 1. That leaves
\[ \operatorname{Stab}(1) = { e, f } \tag{5} \]
which is a subgroup, in fact one we've seen before, the non-normal subgroup from post 2. The stabilizer is always a subgroup, and it's quick to see: \(e\) fixes \(x\); if \(g\) and \(h\) both fix \(x\) then \((gh) \cdot x = g \cdot (h \cdot x) = g \cdot x = x\); and if \(g \cdot x = x\) then applying \(g^{-1}\) to both sides gives \(x = g^{-1} \cdot x\). The orbit has 3 elements, the stabiliser has 2, and
\[ 3 \times 2 = 6 = |D_3| \tag{6} \]
1.3 Orbit-stabilizer
This follows from Lagrange’s Theorem.
Claim: the cosets of \(\operatorname{Stab}(x)\) correspond exactly to the points of \(\operatorname{Orb}(x)\). Map the coset \(g\operatorname{Stab}(x)\) to the point \(g \cdot x\), and check the three things.
Well defined. Replace \(g\) by \(gh\) with \(h \in \operatorname{Stab}(x)\). Then \((gh) \cdot x = g \cdot (h \cdot x) = g \cdot x\), the same point. So the coset determines the answer regardless of which representative we chose.
Injective. Suppose \(g_1 \cdot x = g_2 \cdot x\). Applying \(g_2^{-1}\) gives \((g_2^{-1}g_1) \cdot x = x\), so \(g_2^{-1}g_1 \in \operatorname{Stab}(x)\), which is exactly the condition for \(g_1\) and \(g_2\) to lie in the same coset.
Surjective. Every point of the orbit is \(g \cdot x\) for some \(g\), by definition.
So the number of orbit points equals the number of cosets, which Lagrange tells us is \(|G| / |\operatorname{Stab}(x)|\). Rearranged:
\[ |\operatorname{Orb}(x)| \cdot |\operatorname{Stab}(x)| = |G| \tag{7} \]
This is the orbit-stabilizer theorem. Among other things it says every orbit size divides the group order, which is a strong constraint on how a group can possibly act on anything. The stabilizers of the other two corners are \(\operatorname{Stab}(2) = {e, r^2f}\) and \(\operatorname{Stab}(3) = {e, rf}\) They're related: in post 3 we computed \(rfr^{-1} = r^2f\), which says
\[ r \operatorname{Stab}(1) r^{-1} = \operatorname{Stab}(2) \tag{8} \]
Stabilizers of points in the same orbit are conjugate.
1.4 A group acting on itself
A group can also act on its own elements, and the interesting way to do it is by conjugation:
\[ g \cdot x = gxg^{-1} \tag{9} \]
Check it's an action: \(e \cdot x = x\) clearly, and
\[ (gh) \cdot x = (gh)x(gh)^{-1} = g(hxh^{-1})g^{-1} = g \cdot (h \cdot x) \tag{10} \]
So it qualifies. Now everything above applies, and the orbits are called the conjugacy classes. Let's find them in \(D_3\). The identity is alone, since \(geg^{-1} = e\) always. For \(r\), we already know \(frf^{-1} = r^2\) from post 3, and rotations commute with each other, so the class of \(r\) is \({r, r^2}\). For \(f\), we have \(rfr^{-1} = r^2f\), and computing the other conjugate gives \(r^2 f (r^2)^{-1} = rf\), so all three reflections are in one class. The classes are
\[ { e }, \qquad { r, r^2 }, \qquad { f, rf, r^2f } \tag{11} \]
with sizes 1, 2 and 3. They partition the group, as orbits must, and each size divides 6, as orbit-stabilizer requires.
1.5 Normality
A subgroup \(N\) is normal when \(gNg^{-1} = N\) for every \(g\), which says that \(N\) is closed under the conjugation action. In other words, a subgroup is normal exactly when it's a union of conjugacy classes. For the rotations:
\[ { e, r, r^2 } = { e } \cup { r, r^2 } \tag{12} \]
a union of whole classes, so normal. And the reflections subgroup:
\[ { e, f } = { e } \cup \text{part of} { f, rf, r^2f } \tag{13} \]
which cuts a class in half, so not normal. Both results we derived by going through conjugations in post 3. One last thing; elements whose conjugacy class is just themselves are the ones commuting with everything, and they form the centre of the group. In \(D_3\) only \(e\) has a class of size one, so the centre is trivial (there's no symmetry of the triangle that commutes with all the others).
1.6 Where this is going
So far our sets have been finite and structureless: three corners, six group elements. The natural next move is to let a group act on something with more structure. Specifically, a vector space, where the permutations become invertible matrices. That's called a representation. In fact, every physical statement about spin, about particles, about which fields can exist, is a statement about representations of a particular group.