Abstract Algebra - Group Theory 1

Binary Operation

• A binary operation is a function ⋆ defined by ⋆: GxG → G.
• For any elements a, b in the set G, the operation is denoted as a ⋆ b = ⋆(a,b).

Associativity

• A binary operation is associative if it satisfies the condition: a ⋆ (b ⋆ c) = (a ⋆ b) ⋆ c for all a, b, c in G.

Commutativity

• Two elements a and b in G commute if a ⋆ b = b ⋆ a.
• A binary operation is commutative if this condition holds for all elements in G.

Groups

• A group consists of an ordered pair (G, ⋆) where G is a set and ⋆ is a binary operation.
• Groups must satisfy three axioms:
  • Associativity
  • Existence of an identity element e such that e ⋆ a = a ⋆ e = a for all a in G.
  • Existence of an inverse a⁻¹ for each a such that a ⋆ a⁻¹ = a⁻¹ ⋆ a = e.

Abelian Group

• An abelian group is one where the binary operation is commutative: a ⋆ b = b ⋆ a for all a, b in G.

Finite Group

• A finite group contains a finite number of elements.

Propositions for Groups (Part 1)

• The identity element in a group is unique.
• Each element has a unique inverse.
• The inverse of the inverse: (a⁻¹)⁻¹ = a.
• The inverse of a product: (a ⋆ b)⁻¹ = b⁻¹ ⋆ a⁻¹.
• The product of elements a₁, a₂, ..., aₙ is independent of parentheses (generalized associativity).

Propositions for Groups (Part 2)

• If au = av, then it must follow that u = v.
• If ua = va, then u = v holds.

Order

• The order of an element x in G is the smallest positive integer n such that xⁿ = 1, denoted as |x|.
• If no such integer n exists, x is categorized as having infinite order.

Dihedral Group

• Denoted as Dₙ, representing the symmetries of an n-gon.
• It consists of two types of elements:
  • r: rotation by 2π/n radians.
  • s: reflection.
• The total number of elements in Dₙ is 2n, making its order 2n.

Symmetric Group

• The symmetric group is the group of permutations of a set.

Homomorphism

• A homomorphism is a map f: G → H where f(xy) = f(x)f(y) for all x, y in G.

Isomorphism

• An isomorphism is a bijective homomorphism f: G → H indicating that G and H have the same algebraic structure.

Non-Isomorphic Groups

• Groups G and H are non-isomorphic if:
  • They have different sizes (|G| ≠ |H|).
  • One is abelian while the other is not.
  • For any x in G, the orders differ (|x| ≠ |f(x)|).
• Disproving any one of these statements establishes non-isomorphism.

Group Actions

• A group action of G on a set A is a mapping from G x A to A, denoted as g * a.
• Group actions adhere to two core properties:
  • g₁ * (g₂ * a) = (g₁g₂) * a.
  • The identity element acts as: 1 * a = a for all a in A.