## 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?)

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?