## Groups and Group Actions: Lecture 3

In which we start to explore permutations.

• Proposition 6: A Cayley table is a Latin square: each element appears exactly once in each row and in each column.  For a fixed element $g$ of $G$, we considered row $g$ by defining a map $f_g: G \to G$ via $f_g(g') = gg'$, and noted that this is a bijection.
• Definition of a subgroup of a group.  (We write $H \leq G$ to mean that $H$ is a subgroup of $G$.)
• Definition of the order of an element of a group, and of what it means for an element of a group to have infinite order.
• Definition of an isomorphism between two groups, and of what it means for two groups to be isomorphic.
• Definition of a permutation, and of the set $\mathrm{Sym}(S)$ for a set $S$.  We noted that in this course we shall write permutations on the right.
• Theorem 7: Let $S$ be a set.
• $\mathrm{Sym}(S)$ is a group under composition, called the symmetry group of $S$.
• If $|S| \geq 3$, then $\mathrm{Sym}(S)$ is non-Abelian.
• We have $|S_n| = n!$.

The first part is a standard check. We showed that the group is non-Abelian by explicitly exhibiting two permutations in $\mathrm{Sym}(S)$ that do not commute.  To count permutations, we considered the number of possibilities for where each of $1$, $2$, …, $n$ is sent.

• Definitions of a cycle and a $k$-cycle and the length of a cycle and a transposition.
• Definition of what it means for two cycles to be disjoint.
• Proposition 8: Let $\alpha = (a_1 \dotsc a_k)$ and $\beta = (b_1 \dotsc b_{\ell})$ be disjoint cycles.  Then $\alpha$ and $\beta$ commute.  We checked this by working out where each number is sent to by $\alpha \beta$ and $\beta \alpha$.

### Understanding today’s lecture

We’ve already seen some examples of groups.  Can you come up with some interesting subgroups?  Can you find a subgroup of a finite group?  An infinite subgroup of an infinite group?  A finite subgroup of an infinite group?

Can you find the orders of the elements of $D_6$?  Can you find an example of a group with an element of infinite order?  Can you find an example of a group with an element of infinite order and with another element of finite order?

Pick an explicit example of a permutation in $S_n$.  Can you write it using cycle notation?  Can you write it as a product of disjoint cycles?  In how many ways can you write it as a product of disjoint cycles?

We’ve seen lots of definitions so far in the course; you might enjoy reading the musings of Tim Gowers on what definitions are.  I also recommend his blog post about alternative definitions, which links nicely with my comment in the lecture today when we showed that $f_g$ (in Proposition 6) is a bijection by writing down an inverse.  Some of the examples he discusses are ones that you’ve already met in Analysis and Linear Algebra, others are things you’ll meet later in the Groups course (so you’ll get lots out of reading a little ahead via his blog post).

There are lots of interesting points in this blog post by Tim Gowers — but do be careful, he writes permutations on the left, whereas we write permutations on the right.

### Preparation for Lecture 4

If I give you a permutation written as a product of disjoint cycles, how can you work out its order?  What information do you need?

We’ll use the first part of Sheet 2 Q3 to help us prove a result, so you might find it helpful to have looked at that particular question before the lecture.

We’ll think a bit about permutation matrices, which you touched on briefly in the Linear Algebra II course (when you thought about determinants) — it would be worth reviewing that part of the Linear Algebra course before the lecture.

Pick any cycle (of any length).  Can you write it as a product of transpositions?