Group Theory
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1. Group Theory
2. Applications
1. Group Theory
⑴ Axioms
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: 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: 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
Figure 1. The major algebraic systems
① Set: A collection of elements. No operation/rule
② Magma: One operation (*). For any a, b, a*b is always defined (closed set)
③ Semigroup: One operation (*). Magma + associativity
④ Monoid: One operation (*). Semigroup + existence of identity element e (e*a = a*e = a)
⑤ Group: One operation (*). Monoid + every element has an inverse
○ Abelian group: Group + commutativity $a*b = b*a$
○ In the Abelian case, the multiplicative notation $xy$ is often replaced by additive notation $x+y$.
○ Product group: Defined as the Cartesian product $G \times H$ of two groups $G$, $H$ (e.g., ($g_1$, $h_1$))
○ The new group operation is $(g_1, h_1)\cdot(g_2, h_2) = (g_1g_2, h_1h_2)$
○ Symmetric group of degree $n$: The group of all permutations of $n$ elements. Denoted by $S_n$. Related to the definition of symmetry.
○ The identity element provides a reference point, and inverses guarantee cancellation/solvability, making a group a structure in which computations are possible.
○ Cayley’s theorem: It shows that from the abstract point of view the theory of groups is coextensive with the theory of transformation groups.
⑥ Ring: Two operations +, ·
○ Condition 1. (R, +) is an Abelian group
○ Condition 2. · is associative
○ Condition 3. Distributive laws a(b + c) = ab + ac, (a+b)c = ac + bc
○ Types of rings
○ Unital ring: A ring with multiplicative identity 1
○ Commutative ring: A ring in which ab = ba holds
○ Subring: A non-empty subset $S$ of a ring $R$ such that, when the operations defined on $R$ are restricted to $S$, the elements of $S$ form a ring.
○ Cosets: Cosets are non-empty and disjoint, and whose union is the full ring $R$.
○ Elements of rings
○ If an element $x$ in $R$ has an inverse, then $x$ is said to be regular / invertible / non-singular.
○ Elements which are not regular are called singular.
○ Every ring with non-zero elements has at least two distinct ideals.
⑦ Division ring(skew field): Ring + division possible
○ A ring with identity is called a division ring if every nonzero element has a multiplicative inverse (multiplication need not be commutative)
⑧ Field: Division ring + multiplicative commutativity (i.e., division by any nonzero element + commutative)
○ Subfield: A subset contained in a field that remains a field under the same operations
○ Field extension: When K ⊆ L and both are fields, refers to L / K
⑨ Integral domain: Commutative ring(usually with 1) + no zero divisors (ab = 0 ⇒ a = 0 or b = 0)
⑩ 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 (linear space): A module whose scalar ring is a field; i.e., a module over a field
⑫ 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: 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: Two operations (∧, ∨). Has meet/join operations like gcd/lcm, and satisfies commutative/associative/absorption laws
⑶ Mapping
① Injective function(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(onto)
○ A function whose codomain = image
③ Bijective function
○ A function that is both injective and surjective
④ Homomorphism
○ Given two groups (G, ·), (H, *), a map ρ: G → H satisfying ρ(u · v) = ρ(u) * ρ(v)
⑤ Isomorphism
○ A homomorphism satisfying a one-to-one correspondence (bijective)
○ Two groups that are isomorphic have the same number of elements and the same group structure, differing only in nonessential aspects such as notation and terminology.
⑥ Automorphism
⑦ 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 (= Tg(x))
⑧ Representation (structure theory)
○ 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
○ Purpose: To reconstruct the entire structure of a group $G$ by using many homomorphic images of $G$ (e.g., building a 3D object from its 2D cross-sections).
○ All possible homomorphic images of $R$ can be constructed by means of the ideals in $R$.
○ Reducible: When a representation can be decomposed into a direct sum of other representations
○ Irreducible: Any representation that is not reducible. The set of irreducible representations is denoted Irr(G), irreps, etc.
○ Finite Commutative Group
○ All irreducible representations of a 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: 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,
⑸ Theorem 2. If $R$ is a commutative ring with identity,
① 2-1. $R$ is a field. That is, it has non-trivial ideals.
② 2-2. An ideal $I$ in $R$ is maximal. That is, $R/I$ is a field.
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 The reason general quintic equations cannot be solved by radicals (e.g., √) is that the symmetry of the roots—captured by the Galois group—has a structure that cannot be built using only radical extensions.
⑺ Proof of Fermat’s Last Theorem
⑻ Noether’s Theorem
① If a physical system (its Lagrangian/action) has a continuous symmetry, a corresponding conserved quantity exists.
② It is most naturally formulated for continuous groups (Lie groups).
③ Time translation symmetry (ℝ under addition) → conservation of energy.
④ Spatial translation symmetry (ℝ³) → conservation of momentum.
⑤ Rotational symmetry (SO(3)) → conservation of angular momentum.
⑥ Gauge-group symmetry (e.g., U(1), SU(2), SU(3)) → conservation of charge/current (and related conservation laws in field theory).
Input: 2021.03.27 22:54
Revision: 2026.01.02 15:42