Home Online Mathematical Communication Theorem of the week Groups and Group Actions: Lecture 13

Groups and Group Actions: Lecture 13

Theorem of the week 2018-03-12

In which we explore the Orbit-Stabiliser Theorem.

  • Proposition 58: Let G be a group acting on a set X.  Take x, y \in X with x and y lying in the same orbit.  Then \mathrm{Stab}(x) and \mathrm{Stab}(y) are conjugate: there is g \in G with \mathrm{Stab}(x) = g^{-1} \mathrm{Stab}(y) g.  We noted that since x and y are in the same orbit, there is g \in G with g \cdot x = y.  And then we showed that h \in \mathrm{Stab}(x) if and only if h \in g^{-1} \mathrm{Stab}(y) g.
  • Theorem 59 (Orbit-Stabiliser Theorem): Let G be a finite group acting on a set X.  Take x \in X.  Then |G| = |\mathrm{Orb}(x)| \cdot |\mathrm{Stab}(x)|.  We defined a map from G/\mathrm{Stab}(x) to \mathrm{Orb}(x) and showed that it’s a bijection, then used Lagrange’s theorem.
  • Corollary 60: Let G be a finite group, take g \in G.  Then |G| = |C_G(g)| \cdot |ccl_G(g)|, where C_G(g) = \{h \in G : gh = hg \} is the centraliser of g in G and ccl_G(g) = \{hgh^{-1} : h \in G\} is the conjugacy class of g in G.  We have already seen that G acts on itself via conjugation, and that for g \in G we have \mathrm{Stab}(g) = C_G(g) and \mathrm{Orb}(x) = ccl_G(g), so the result follows immediately from Orbit-Stabiliser.
  • Proposition 61: Let p be prime.  Let G be a group with order p^r for some r \geq 1.  Then the centre of G is non-trivial.  We used the action of G on itself via conjugation.  The elements of the centre Z(G) are precisely the elements whose orbits have size 1.  The size of each orbit divides |G| and so is 1 or a multiple of p, and since the orbits partition G we see that the sum of their sizes is a multiple of p.  So the number of orbits of size 1 is a multiple of p, and since it’s at least 1 it must be at least p.

Understanding today’s lecture

By thinking of the group of rotational symmetries of the cube as acting on the edges of the cube, can you show that the group has size 24?  What is the size of the group of rotational symmetries of the tetrahedron?  Of the dodecahedron?

There’s a hedgehogmaths video in which I talk through the statement and proof of the Orbit-Stabiliser theorem.

In true Blue Peter style, here’s a cube I made earlier (using instructions by beAd Infinitum).

Beaded bead - cube

Here’s a complete set of beaded Platonic solids made using the same instructions.

Beaded_Platonic_solids

And here’s a dodecahedron I made earlier (using instructions from NRICH), in case you’re having difficulties visualising one.20150511_115247-1

Here is an exciting animation of a spinning dodecahedron.  If you want to make your own dodecahedron and don’t feel like origami or beading, then print a net and get some glue.

Further reading

Here’s Tim Gowers on the orbit-stabiliser theorem.  And there are some useful summary notes here from Thomas Beatty.  These lecture notes by Keith Carne show how one can start with orbit-stabiliser and get to all sorts of exciting mathematics.

Preparation for Lecture 14

What can you say about the centre of a group of order p^2 (where p is prime)?

What can you say about the structure of a group of order p^2?