## Groups and Group Actions: Lecture 13

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).

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

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

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.

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 $p^2$?