Group Theory

Introduction to Group Theory

Group Axioms: a set \mathbb{G} of elements \{X, Y, Z, ...\} with an operation \cdot such that for all X, Y, Z \in \mathbb{G}:

Note that it is not a requirement for \cdot to be commutative. In fact, most of interesting groups are non-commutative such as 3D rotation (while 2D rotation is commutative).

The set of group G can be thought as a set of actions where actions are defined in relation to each other in the consideration of symmetry.

Group Actions: a group \mathbb{G} can act on another set V to transform v \in V. For X, Y, E \in \mathbb{G} and the action \cdot, it must be that:

For intuitive understanding of Group and its relation to String Theory, refer to 3b1b's Video

Isomorphism: Two groups that shares symmetry. For example, applying 3 times of 8 distinct permutation on 4 objects is the same as 120 or 140 degree rotation about each diagonal on a cube.


To categorize all finite groups, we do two things:

  1. Find all the simple groups: there are 18 distinct infinite families of simple groups plus 26 sporadic groups (which contains 20 groups in happy family where the Monster Group and Baby Monster Group, and 6 Pariahs).
  2. Find all the ways to combine simple groups

Lie Group


Lie Group (formally known as "Continuous Transformation Groups): a group that is also a smooth (no edges or spikes) manifold (3D surface) such that the elements of the manifold satisfy the group axioms.


Complex Number Review

Unit Complex Number (complex number on unit circle S^1):

U = \{z | z^* \cdot z = 1\}

Observe that

(\forall z \in U)(z = \cos \theta + i \sin \theta \implies z \text{ is a rotation operator of degree } \theta \text{ when multiplied with a complex number})

You can easily compose rotational operator by doing z_3 = z_1 \cdot z_2.

For more details, see my article on Complex_Number

SO(2): The 2D Rotation Matrices

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