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