Abstract Algebra - Group Theory 1

Study Notes

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.

Description

Test your knowledge of key concepts in group theory with these flashcards. Learn about binary operations, associativity, and commutativity, essential building blocks in abstract algebra. This quiz will help reinforce your understanding of the foundational principles in group theory.