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抽象代数代考| Groups, the first glance with an example of symmetries of a triangle

An informal definition of groups

A group is a set of actions, satisfying a few mild properties.

Proposition 1.

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:

Different minimal generating sets might lead to different diagrams
Remark 1.

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”:

The symmetry group of the triangle is generated by just two: Tri $=\langle r, f\rangle .$

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.

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