Group Theory
Recommended Reading: 【Mathematics】 Mathematics Table of Contents
1. Group Theory
2. Functions
1. Group Theory
⑴ Axiom
① Closure: ∀a, b ∈ G, a·b ∈ G
② Associative Property: ∀a, b, c ∈ G, (a·b)·c = a·(b·c)
③ Identity Element: For any a ∈ G, there exists e ∈ G such that a·e = e·a = a
④ Inverse Element: For any a ∈ G, there exists a-1 such that a·a-1 = a-1·a = e
○ Uniqueness of Inverse: For any two inverses a0-1, a1-1, a0-1 = a1-1 always holds
○ a0-1 = e·a0-1 = (a1-1·a)·a0-1 = a1-1·(a·a0-1) = a1-1·e = a1-1
⑵ Terminology
① Homomorphism: For two groups (G, ·), (H, *), a map ρ such that ρ(u · v) = ρ(u) * ρ(v), ρ: G → H
② Isomorphism: A homomorphism that satisfies bijective
③ Product Group: Defined as the Cartesian product G × H from two groups G, H (e.g., (g1, h1))
○ New group operation is defined as (g1, h1)·(g2, h2) = (g1g2, h1h2)
④ Commutativity: A group (G, +) is commutative or Abelian if the operation order does not matter, i.e., a·b = b·a
○ If this property does not hold for all elements of the group, the group is non-commutative
○ Finite commutative groups can be expressed as products of cyclic groups
⑤ Group Action: Defined as a function T: G × X → X that maps a pair (g, x) to an element in X
○ Condition 1. For the identity element e of G, T(e, x) = x
○ Condition 2. For g1, g2 ∈ G, T(g1, T(g2, x)) = T(g1g2, x)
○ Simplified notation: Represent T(g, x) as Tg(x), meaning point x is mapped to gx (= Tg(x))
⑥ Invariance: A function ϕ: X → Y has G-invariance if for all g ∈ G and x ∈ X, ϕ(x) = ϕ(gx)
○ This implies that the group action on the input space has no effect on the output
⑦ Equivariance: A function ϕ: X → Y has G-equivariance if for all g ∈ G and x ∈ X, ϕ(gx) = g’ϕ(x)
○ Here, g’ ∈ G’ is homomorphic to G and acts on the output space
○ Equivariance implies that the group action on the input space induces a corresponding group action on the output space
⑧ Orbit: For a point x ∈ X, the orbit Gx is defined as the set {gx: g ∈ G}
○ For example, if G is a translation group and x is an image, the orbit represents all shifted versions of the image
⑨ Homogeneous Space: A space X where for any pair x1, x2 ∈ X, there exists g ∈ G such that gx1 = x2
○ A homogeneous space X with a group action G is also called a G-space
⑩ Representation: A representation of a group G is a map ρ: G → GL(V), where GL(V) is the group of linear transformations on vector space V
○ Homomorphism condition: Representations must satisfy ρ(g1g2) = ρ(g1)ρ(g2)
○ In many cases, V is ℝn or ℂn
○ Reducible: If the representation can be decomposed into a direct sum of other representations
○ Irreducible: All representations that are not reducible. The set of irreducible representations is denoted Irr(G), irreps, etc.
○ Irreducible representations of finite commutative groups are all one-dimensional and are in bijection with the group itself
○ Irreducible representations of finite commutative groups are classified as products of cyclic groups ℤ/nℤ
⑶ Theorem 1. Each axiom of the group 〈G, *〉 is a structural property of 〈G, *〉
① Definition: Structural property of 〈G, *〉 refers to properties shared by all binary structures isomorphic to 〈G, *〉
② Premise: Through isomorphism φ: G → S, 〈S, ㆍ〉 is a binary structure isomorphic to 〈G, *〉
③ Proof for Axiom 1: For α, β, γ ∈ S, ∃ a, b, c ∈ G such that φ(a) = α, φ(b) = β, φ(c) = γ
④ Proof for Axiom 2
⑤ Proof for Axiom 3: For the inverse a’ of a ∈ G,
2. Functions
⑴ Injective Function: A function f: X → Y where each element in the range corresponds to exactly one element in the domain
⑵ Surjective Function: A function where the codomain equals the range
⑶ Bijective Function: A function that is both injective and surjective
Input: 2021.03.27 22:54
Revision: 2024.12.21 15:42