## Groups and Group Actions: Lecture 14

In which we explore groups of order $p^2$ and encounter Cauchy’s Theorem.

• Lemma 62: Let $p$ be a prime, let $G$ be a group of order $p^2$.  Then $G$ is Abelian.  We saw last time that the centre of $G$, $Z(G)$, is non-trivial, so in this case it must have order $p^2$ (and so be the whole group, making $G$ Abelian) or $p$.  In the latter case, we saw that $G/Z(G)$ is cyclic and used this to deduce that in fact $G$ is Abelian.
• Proposition 63: Let $p$ be a prime, let $G$ be a group of order $p^2$.  Then $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$.  In the case that $G$ contains no element of order $p^2$, we picked suitable $x$ and $y$ and used the fact that $G$ is Abelian to show that $G \cong \langle x \rangle \times \langle y \rangle$.
• Theorem 64 (Cauchy‘s Theorem): Let $G$ be a finite group, let $p$ be a prime dividing $|G|$.  Then $G$ contains an element of order $p$.  We considered the set $S = \{ (g_1, g_2, \dotsc, g_p) \in G^p : g_1 g_2 \dotsm g_p = e \}$, and the action on it by $H = \langle \sigma \rangle$ where $\sigma = (1\,2\,\dotsc\,p)$ defined by $\sigma \cdot (g_1, g_2, \dotsc, g_p) = (g_2, g_3, \dotsc, g_p, g_1)$.  By considering the sizes of the orbits, we found a non-trivial orbit of size 1 (not just $\{ (e,e,\dotsc,e)\}$), and showed that this corresponded to an element $g \in G$ of order $p$.

### Understanding today’s lecture

Would the arguments in Lemma 62 and Proposition 63 work for groups of order $p^3$?  Is every group of order $p^3$ Abelian?

What are the groups with exactly two conjugacy classes?

What goes wrong in our proof of Cauchy’s Theorem if $p$ is not prime?  (We know that the result isn’t true in general, don’t we?  Can you give a group of order 4 that contains no element of order 4?)

As I mentioned, there are several results known as Cauchy’s Theorem.

I mentioned the structure theorem for finite Abelian groups, which is included in the Algebra 2 (Rings and Modules) course.

There are other ways to prove Cauchy’s Theorem, here is a poster for your bedroom wall.

### Preparation for Lecture 15

On Sheet 6, there was a question about counting orbits of colourings of the edges of a triangle under the action of $D_6$, using first two and then three colours.  Can you generalise to $n$ colours?

What is the group of symmetries of the cube?  We talked about the size of the group of rotational symmetries, what is the size of the group of all symmetries?  What is the structure of the group?