Binary Operations, Monoids, Groups and Subgroups

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✓ 15 results · all verified · 13 also independently AI-judged

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1 · Prerequisites

Real Analysis

Set Theory

2 · Logical flowchart

3 · Items

Definition

Binary operation on a set; associativity, commutativity, and a subset closed under the operation

Definition

Left identity, right identity, and two-sided identity for a binary operation

Lemma

A left identity and a right identity for the same binary operation are equal; hence there is at most one two-sided identity

Definition

Semigroup and monoid

Definition

Left inverse, right inverse, and invertible element of a monoid

Lemma

In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided

Definition

Group and abelian group

Lemma

In a group e⁻¹ = e, (g⁻¹)⁻¹ = g and (gh)⁻¹ = h⁻¹g⁻¹, the order of the last product being essential

Lemma

Cancellation in a group: gx = gy or xg = yg forces x = y; equivalently left and right translation by g are bijections of G, so gx = h and xg = h each have exactly one solution

Lemma

The product g₀ g₁ ⋯ gₙ₋₁ of a finite list in a monoid, by recursion, with the empty product (n = 0) equal to the identity

Theorem

Generalised associativity: in a monoid the product of a finite list does not depend on the bracketing, and in a commutative monoid it does not depend on the order of the factors either

Definition

One-step subgroup test: a nonempty H ⊆ G is a subgroup iff gh⁻¹ ∈ H for all g, h ∈ H; the identity and the inverses of H are then those of G

Lemma

The order |G| of a finite group and the order ord(g) of an element, with ord(g) = ∞ when no positive power of g is the identity

Lemma

If ord(g) = n then gᵏ = e iff k is an integer multiple of n, the powers g⁰, …, gⁿ⁻¹ are distinct, and ⟨ g ⟩ has exactly n elements; if g has infinite order then gʲ = gᵏ only for j = k

Definition

Monoid homomorphism and group homomorphism

Lemma

A group homomorphism automatically satisfies f(e) = e' and f(g⁻¹) = f(g)⁻¹, and f(gⁿ) = f(g)ⁿ for every n ∈ ℤ; for monoid homomorphisms preservation of the identity must be assumed

Binary Operations, Monoids, Groups and Subgroups

3 · Items

Binary operation on a set; associativity, commutativity, and a subset closed under the operation

Left identity, right identity, and two-sided identity for a binary operation

A left identity and a right identity for the same binary operation are equal; hence there is at most one two-sided identity

Semigroup and monoid

Left inverse, right inverse, and invertible element of a monoid

In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided

Group and abelian group

In a group e⁻¹ = e, (g⁻¹)⁻¹ = g and (gh)⁻¹ = h⁻¹g⁻¹, the order of the last product being essential

Cancellation in a group: gx = gy or xg = yg forces x = y; equivalently left and right translation by g are bijections of G, so gx = h and xg = h each have exactly one solution

The invertible elements of a monoid form a group under the restricted operation

The symmetric group Sym(X): the bijections of a set X under composition

Sym(X) is a group under composition, and it is non-abelian whenever X has at least three distinct elements

The product g₀ g₁ ⋯ gₙ₋₁ of a finite list in a monoid, by recursion, with the empty product (n = 0) equal to the identity

Generalised associativity: in a monoid the product of a finite list does not depend on the bracketing, and in a commutative monoid it does not depend on the order of the factors either

Powers gⁿ: natural exponents in a monoid and integer exponents in a group, with g⁰ = e

Exponent laws in a group: gᵐ⁺ⁿ = gᵐgⁿ and (gᵐ)ⁿ = gᵐⁿ for all m, n ∈ ℤ, and (gh)ⁿ = gⁿhⁿ when g and h commute

Subgroup

One-step subgroup test: a nonempty H ⊆ G is a subgroup iff gh⁻¹ ∈ H for all g, h ∈ H; the identity and the inverses of H are then those of G

The intersection of a nonempty family of subgroups of G is a subgroup of G

The subgroup ⟨ S ⟩ generated by a subset, the cyclic subgroup ⟨ g ⟩, and cyclic groups

⟨ g ⟩ = { gⁿ : n ∈ ℤ }, and every cyclic group is abelian

Division with remainder in ℤ: for a ∈ ℤ and b > 0 there are unique q, r ∈ ℤ with a = qb + r and 0 ≤ r < b

In a finite group, every element g satisfies gⁿ = e for some natural n ≥ 1

The order |G| of a finite group and the order ord(g) of an element, with ord(g) = ∞ when no positive power of g is the identity

If ord(g) = n then gᵏ = e iff k is an integer multiple of n, the powers g⁰, …, gⁿ⁻¹ are distinct, and ⟨ g ⟩ has exactly n elements; if g has infinite order then gʲ = gᵏ only for j = k

Monoid homomorphism and group homomorphism

A group homomorphism automatically satisfies f(e) = e' and f(g⁻¹) = f(g)⁻¹, and f(gⁿ) = f(g)ⁿ for every n ∈ ℤ; for monoid homomorphisms preservation of the identity must be assumed

Binary Operations, Monoids, Groups and Subgroups

3 · Items

Binary operation on a set; associativity, commutativity, and a subset closed under the operation

Left identity, right identity, and two-sided identity for a binary operation

A left identity and a right identity for the same binary operation are equal; hence there is at most one two-sided identity

Semigroup and monoid

Left inverse, right inverse, and invertible element of a monoid

In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided

Group and abelian group

In a group e⁻¹ = e, (g⁻¹)⁻¹ = g and (gh)⁻¹ = h⁻¹g⁻¹, the order of the last product being essential

Cancellation in a group: gx = gy or xg = yg forces x = y; equivalently left and right translation by g are bijections of G, so gx = h and xg = h each have exactly one solution

The invertible elements of a monoid form a group under the restricted operation

The symmetric group Sym(X): the bijections of a set X under composition

Sym(X) is a group under composition, and it is non-abelian whenever X has at least three distinct elements

The product g₀ g₁ ⋯ gₙ₋₁ of a finite list in a monoid, by recursion, with the empty product (n = 0) equal to the identity

Generalised associativity: in a monoid the product of a finite list does not depend on the bracketing, and in a commutative monoid it does not depend on the order of the factors either

Powers gⁿ: natural exponents in a monoid and integer exponents in a group, with g⁰ = e

Exponent laws in a group: gᵐ⁺ⁿ = gᵐgⁿ and (gᵐ)ⁿ = gᵐⁿ for all m, n ∈ ℤ, and (gh)ⁿ = gⁿhⁿ **when g and h commute**

Subgroup

One-step subgroup test: a nonempty H ⊆ G is a subgroup iff gh⁻¹ ∈ H for all g, h ∈ H; the identity and the inverses of H are then those of G

The intersection of a nonempty family of subgroups of G is a subgroup of G

The subgroup ⟨ S ⟩ generated by a subset, the cyclic subgroup ⟨ g ⟩, and cyclic groups

⟨ g ⟩ = { gⁿ : n ∈ ℤ }, and every cyclic group is abelian

Division with remainder in ℤ: for a ∈ ℤ and b > 0 there are unique q, r ∈ ℤ with a = qb + r and 0 ≤ r < b

In a finite group, every element g satisfies gⁿ = e for some natural n ≥ 1

The order |G| of a finite group and the order ord(g) of an element, with ord(g) = ∞ when no positive power of g is the identity

If ord(g) = n then gᵏ = e iff k is an integer multiple of n, the powers g⁰, …, gⁿ⁻¹ are distinct, and ⟨ g ⟩ has exactly n elements; if g has infinite order then gʲ = gᵏ only for j = k

Monoid homomorphism and group homomorphism

A group homomorphism automatically satisfies f(e) = e' and f(g⁻¹) = f(g)⁻¹, and f(gⁿ) = f(g)ⁿ for every n ∈ ℤ; for monoid homomorphisms preservation of the identity must be assumed

Exponent laws in a group: gᵐ⁺ⁿ = gᵐgⁿ and (gᵐ)ⁿ = gᵐⁿ for all m, n ∈ ℤ, and (gh)ⁿ = gⁿhⁿ when g and h commute