Group Theory

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1. Group Theory

2. Applications

1. Group Theory

⑴ Axioms(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 satisfying a·e = e·a = a exists

④ Inverse element: For any a ∈ G, there exists a^-1 such that a·a^-1 = a^-1·a = e

○ Uniqueness of the inverse: For any two inverses a₀^-1, a₁^-1, we always have a₀^-1 = a₁^-1

○ a₀^-1 = e·a₀^-1 = (a₁^-1·a)·a₀^-1 = a₁^-1·(a·a₀^-1) = a₁^-1·e = a₁^-1

⑤ Commutativity: When the order of the operation does not matter for the group (G, +), i.e., when a·b = b·a

○ Called commutative or Abelian group.

○ If this property does not hold for all elements of a group, the group is called a non-commutative group

○ A finite commutative group can be expressed as a product of cyclic groups(cyclic group)

⑥ Invariance(invariance): For a function ϕ: X → Y to be G-invariant, for all g ∈ G and x ∈ X, it must satisfy ϕ(x) = ϕ(gx)

○ This means the group action on the input space has no effect on the output

⑦ Equivariance(equivariance): For a function ϕ: X → Y to be G-equivariant, for all g ∈ G and x ∈ X, it must satisfy ϕ(gx) = g’ϕ(x)

○ Here g’ ∈ G’ is a group that is homomorphic to G and acts on the output space

○ Equivariance means that the group action in the input space induces a corresponding group action in the output space

⑵ Algebraic structures

① Set(set): A collection of elements. No operation/rule

② Magma(magma): One operation (*). For any a, b, a*b is always defined (closed set)

③ Semigroup(semigroup): One operation (*). Magma + associativity

④ Monoid(monoid): One operation (*). Semigroup + existence of identity element e (e*a = a*e = a)

⑤ Group(group): One operation (*). Monoid + every element has an inverse

○ Abelian group(Abelian group): Group + commutativity ab = ba

○ Product group(product group): Defined as the Cartesian product G × H of two groups G, H (e.g., (g₁, h₁))

○ The new group operation is (g₁, h₁)·(g₂, h₂) = (g₁g₂, h₁h₂)

⑥ Ring(ring): Two operations +, ·

○ Definition

○ (R, +) is an Abelian group

○ · is associative

○ Distributive laws a(b + c) = ab + ac, (a+b)c = ac + bc

○ Unital ring(unital ring): A ring with multiplicative identity 1

○ Commutative ring(commutative ring): A ring in which ab = ba holds

⑦ Division ring(division ring, skew field): Ring + division possible

○ Every nonzero element has a multiplicative inverse (multiplication need not be commutative)

⑧ Field(field): Division ring + multiplicative commutativity (i.e., division by any nonzero element + commutative)

○ Subfield(subfield): A subset contained in a field that remains a field under the same operations

○ Field extension(field extension): When K ⊆ L and both are fields, refers to L / K

⑨ Integral domain(integral domain): Commutative ring(usually with 1) + no zero divisors (ab = 0 ⇒ a = 0 or b = 0)

⑩ Module(module): Something like a vector space where a ring R acts as scalars.

○ Definition: (M, +) is an Abelian group, and scalar multiplication R × M → M satisfies distributive/associative rules

⑪ Vector space: A module whose scalar ring is a field; i.e., a module over a field

⑫ Homogeneous space(homogeneous space): A space X such that for any pair x₁, x₂ ∈ X, there always exists g ∈ G with gx₁ = x₂

○ A homogeneous space X equipped with an action of G is also called a G-space(G-space)

⑬ Algebra(algebra): A vector space (or module) over a field (or ring) where multiplication between elements is also defined and is compatible with scalars structure

○ Examples: Matrix algebra, sigma-algebra

⑭ Lattice(lattice): Two operations (∧, ∨). Has meet/join operations like gcd/lcm, and satisfies commutative/associative/absorption laws

⑶ Mapping(mapping)

① Injective function(injective, one-to-one)

○ A function when, in f: X → Y , each element of the codomain is related to exactly one element of the domain

② Surjective function(surjective, onto)

○ A function whose codomain = image

③ Bijective function(bijective)

○ A function that is both injective and surjective

④ Homomorphism(homomorphism)

○ Given two groups (G, ·), (H, *), a map ρ: G → H satisfying ρ(u · v) = ρ(u) * ρ(v)

⑤ Isomorphism(isomorphism)

○ A homomorphism satisfying a one-to-one correspondence(bijective)

⑥ Automorphism(automorphism)

⑦ Group action(group action): A group action is a function T: G × X → X, mapping a pair (g, x) to an element of X

○ Definition: A group (a set of symmetries) acts by permuting solutions (roots) or numbers among each other

○ 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)

○ For brevity, we write T(g, x) as Tg(x), and this is expressed as the point x being mapped to gx (= T_g(x))

⑧ Representation(representation)

○ A representation of a group G means a map ρ: G → GL(V), the group of linear transformations on a vector space V

○ Homomorphism condition: A representation must additionally satisfy ρ(g₁g₂) = ρ(g₁)ρ(g₂)

○ In many cases V is ℝⁿ or ℂⁿ

○ Reducible(reducible): When a representation can be decomposed into a direct sum of other representations

○ Irreducible(irreducible): Any representation that is not reducible. The set of irreducible representations is denoted Irr(G), irreps, etc.

○ All irreducible representations of a finite commutative group(finite commutative group) are 1-dimensional and form a bijection with the group itself correspondence(bijection)

○ Irreducible representations of finite commutative groups are classified as products of cyclic groups ℤ/nℤ

⑨ Orbit(orbit): For a point x ∈ X, the orbit Gx of x is defined as the set {gx: g ∈ G}

○ For example, if G is a translation group and x is an image, the orbit means all translated versions of that image

⑷ Theorem 1. Each axiom of a group 〈G, *〉 is a structural property of 〈G, *〉

① Definition: A structural property of 〈G, *〉 means a characteristic shared by all binary structures isomorphic to 〈G, *〉

② Premise: Via an isomorphism φ: G → S, 〈S, ㆍ〉 is a binary structure isomorphic to 〈G, *〉

③ Proof for Axiom 1: α, β, γ ∈ S, ∃ a, b, c ∈ G s.t. 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. Applications

⑴ In a regular tetrahedron, because the base is an equilateral triangle, the distances that arise within the base are in the √3 family, and the distances that arise when moving into three dimensions are in the √6 family; this can be proved using group theory.

⑵ Irrational number symmetry: For an operation with rational coefficients that yields a + b√c with b ≠ 0, it must also derive a - b√c with b ≠ 0

⑶ $SO(2)$ action and the Lie algebra generator

⑷ A finite cyclic group(Abelian group) is ultimately expressed in terms of cos or sin expressed.

⑸ Inequality proofs and group theory

⑹ Proof of Fermat’s Last Theorem

⑺ Noether’s Theorem