An informal definition of groups
A group is a set of actions, satisfying a few mild properties.
Basic properties
Closure: In any group, composing actions in any order is another action.
Identity: Every group has an identity action $e$, satisfying
$$
a e=e a
$$
for all actions a. [Often we use 1 instead of e.]
Inverses: Every action $a$ in a group has an inverse action $b$, satisfying
$$
a b=e=b a
$$
We will often refer to the the operation of composing actions as multiplication, and will usually write it from left-to-right.
Every group has a generating set, and we will use angle brackets to denote this.
We usually prefer to find a minimal generating set. For example,
$$
\text { Rect }=\langle v, h\rangle=\langle v, r\rangle=\langle h, r\rangle=\langle v, h, r\rangle=\langle v, h, e\rangle=\cdots
$$
There still something missing from the above definition of a group.
Minimal generating sets
We usually only include Cayley diagram arrows corresponding to a minimal generating set.
Different minimal generating sets might lead to different diagrams:
The group Rect has some properties that are not always true for other groups:
it is abelian: $a b=b a$ for all $a, b \in$ Rect
every element is its own inverse: $a^{-1}=a$ for all $a \in$ Rect
the Cayley diagrams for any two minimal generating sets have the same structure
all minimal generating sets have the same size
Symmetries of a triangle
Let’s consider a triangle, and call this its “home state”:
There are six symmetries: three rotations $\left(0^{\circ}, 120^{\circ}, 240^{\circ}\right)$, and three reflections.
The symmetry group of the triangle is generated by just two: Tri $=\langle r, f\rangle$.
- The identity action, 1
- A counterclockwise $120^{\circ}$ rotation, $r$
- A counterclockwise $240^{\circ}$ rotation, $r^{2}$
- A horizontal flip, $f$
- Rotate, then horizontally flip, $r f$
- Rotate twice, then horizontally flip, $r^{2} f$.
Notice that $r f \neq f r$. We say that this group is non-abelian.