The Orbit-Stabilizer Theorem

The other night before turning in, I read Tim Gowers’ post on the orbit-stabilizer theorem. It starts with an invitation to count the rotational symmetries of a cube. He suggests a natural approach: consider the number of ways a rotation could act on a face, on a vertex, on an edge, etc. A face, for instance, can be sent to any one of the six faces. Having chosen one to send it to, there are four places to send one of its corners. Once that’s decided the rotation is determined, so there are altogether 6 \times 4 = 24 rotations.

Similarly, there are eight places to send a vertex. Having chosen one, there are two places to send an adjacent vertex. Once that’s decided, the rotation is determined, so there are altogether 8 \times 3 = 24 rotations.

(A few details are deliberately suppressed in the examples above. In particular, you may want to convince yourself that a rotation followed by a rotation is a rotation and that each sequence of choices discussed above does indeed fix the rotation.)

The general principle is that when a (finite) group G acts on a set X, the number of elements to which x \in X can be sent (the order of its orbit) times the number of elements g \in G that fix x (the order of its stabilizer) is equal to the order of G:

\displaystyle |O_x||S_x|= |G|.

Gowers gives at least four proofs, two of which are reproduced here.

Proof 1. Let y \in X be in the orbit of x \in X and define the set S_{xy} consisting of all g \in G such that gx = y. We show that |S_{xy}| = |S_{x}|. The result follows by letting y run through O_x.

Pick some h \in S_{xy} and define a function \phi_{h} : S_{xy} \to S_{x} (henceforth just \phi) that sends g \in S_{xy} to h^{-1}g. Now consider the function \psi : S_{x} \to S_{xy} which sends u \in S_{x} to hu. One readily verifies that \psi \circ \phi is the identity on S_{xy} and that \phi \circ \psi is the identity on S_{x}. Therefore, since \phi is invertible, |S_{xy}| = |S_{x}| and the result follows. \square

The next proof is apparently quite common.

Proof 2. We show that the elements of O_x are in one-to-one correspondence with the left cosets of S_x. The naive choice \phi which sends gS_x to gx \in O_x works. First, it’s well defined- if gS_x = hS_x, then g = hs for some s \in S_x, so gx = g(sx) = (gs)x = hx. It’s also invertible: let \psi send y = gx \in O_x to gS_x. This is also well-defined- if y = gx = hx then x = g^{-1}hx so that g^{-1}h \in S_x, i.e. gS_x = hS_x. Then \psi \circ \phi is the identity on O_x and \phi \circ \psi is the identity the set of left cosets of S_{x}. Therefore, since \phi is invertible, O_x = [G : S_{x}] = |G|/|S_{x}|.

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