From: 3blue1brown
Group theory is primarily concerned with the study of symmetry [00:01:30].
Understanding Symmetry
When considering a symmetric shape like a square, its symmetry can be defined by all the actions that can be performed on it which leave it looking indistinguishable from its original state [00:01:40].
Examples of such actions on a square include:
- Rotating it 90 degrees counterclockwise [00:01:50].
- Flipping it around its vertical line of symmetry [00:01:57].
Each of these actions is called a “symmetry” of the square [00:02:12].
Groups of Symmetries
All the symmetries of an object, when collected together, form a “group of symmetries” (or simply a “group”) [00:02:16].
The particular group of symmetries for a square consists of 8 distinct symmetries [00:02:27]:
- The action of “doing nothing” (identity) [00:02:30].
- Three different rotations (90°, 180°, 270°) [00:02:34].
- Four different ways to flip the square over (reflections) [00:02:34].
This group of 8 symmetries is specifically known as the dihedral group of order 8 [00:02:39].
Finite vs. Infinite Groups
The dihedral group of order 8 is an example of a finite group, as it contains a limited number of actions (8 actions) [00:02:46].
However, many other groups consist of infinitely many actions [00:02:50]. For instance, the group of all possible rotations of a circle (of any angle) is an infinite group. This group captures all symmetries of the circle that do not involve flipping it [00:02:55]. In this group, every action exists on an infinite continuum between 0 and 2π radians [00:03:06].
Group Arithmetic
A core aspect of group theory is understanding how symmetries interact with each other [00:04:09]. This interaction forms a kind of arithmetic within the group:
- On a square, rotating 90 degrees and then flipping it vertically results in the same overall effect as flipping it over a diagonal line [00:04:15].
- On a circle, rotating 270 degrees followed by a 120-degree rotation is equivalent to a single 30-degree rotation [00:04:35].
In any group, two actions can be “added” or “multiplied” together by applying one after the other to produce a third action within the group [00:05:00]. This collection of underlying relations, defining how pairs of actions compose to form a single equivalent action, is what fundamentally defines a group [00:05:25]. This organization of a collection of actions by their composition relation is crucial to much of modern mathematics [00:05:38].