# 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:

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

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