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 $X, Y, Z \in \mathbb{G}$ :

Composition is in the group: $X \cdot Y \in \mathbb{G}$
Identity is in the group: $(\exists E \in \mathbb{G})(X \cdot E = E \cdot X = X)$
Inverse is in the group: $(\exists X^{-1} \in \mathbb{G})(X^{-1} \cdot X = X \cdot X^{-1} = E)$
Operation is associative: $X \cdot (Y \cdot Z) = (X \cdot Y) \cdot Z$

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:

identity is the null action: $E \cdot v = v$
it is compatible with composition: $(X \cdot Y) \cdot v = X \cdot (Y \cdot v)$

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.

Questions:

What are all the finite groups to isomorphism? (What are all the ways finite things can be symmetric?)
Can groups be broken down to simple groups (Jordan-Holder Theorem)?

To categorize all finite groups, we do two things:

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).
Find all the ways to combine simple groups

Lie Group

Youtube

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.

Manifold

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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