# User:Jshflynn/Hungerford Algebra

## Contents

## About

This is the coverage of the book Algebra by Thomas W. Hungerford.

## $\text{I}$: Groups

### $\S 1$: Semigroups, Monoids and Groups

Definition:Operation/Binary Operation

Definition:Multiplicative Notation

Definition:Product (Abstract Algebra)

Identity is only Idempotent Element in Group/Proof 2

Unique Solution of Group Equation

Equivalence of Axiom Schemata for Groups

Semigroup is Group Iff Latin Square Property Holds

Integers under Addition form Infinite Cyclic Group

Rational Numbers under Addition form Abelian Group

Real Numbers under Addition form Abelian Group

Integers under Multiplication form Countably Infinite Commutative Monoid

Rational Numbers under Multiplication form Commutative Monoid

Non-Zero Rational Numbers under Multiplication form Abelian Group

Non-Zero Real Numbers under Multiplication form Abelian Group

Integer Multiples under Multiplication form Semigroup

Definition:Symmetry Group of Square

Definition:Permutation on n Letters

Definition:Permutation on n Letters/Two-Row Notation

Definition:Group Direct Product

External Direct Product of Groups is Group

External Direct Product of Abelian Groups is Abelian Group

Order of External Direct Product

Definition:Congruence Relation

Definition:Operation Induced on Quotient Set

Quotient Structure of Monoid is Monoid

Quotient Structure of Group is Group

Quotient Structure of Abelian Group is Abelian Group

Congruence (Number Theory) is Congruence Relation

Definition:Additive Group of Integers Modulo m

Order of Additive Group of Integers Modulo m

Reduced Residue System under Multiplication forms Abelian Group

Reduced Residue System under Multiplication forms Abelian Group/Corollary

Definition:Group of Rationals Modulo One

Group of Rationals Modulo One is Group

Definition:Power of Element of Semigroup

Definition:Power of Element/Monoid

Definition:Power of Element/Group

Index Laws for Monoids/Sum of Indices

Index Laws for Monoids/Product of Indices

Natural Numbers under Addition form Commutative Monoid

Multiplicative Semigroup of Integer Multiples/Note

Mappings to Algebraic Structure form Similar Algebraic Structure

Equivalence of Axiom Schemata for Groups/Warning

Definition:Symmetry Group of Square

Reduced Residue System under Multiplication forms Abelian Group/Corollary

Modulo One is Congruence Relation on Rational Numbers

Group of Rationals Modulo One is Group

#### Bookmark

Even Order Group has Order 2 Element

### $\S 2$

Definition:Semigroup Homomorphism

Definition:Semigroup Monomorphism

Definition:Semigroup Epimorphism

Definition:Semigroup Isomorphism

Definition:Semigroup Endomorphism

Definition:Semigroup Automorphism

Composite of Homomorphisms is Homomorphism/Algebraic Structure

Composite of Monomorphisms is Monomorphism

Composite of Epimorphisms is Epimorphism

Composite of Isomorphisms is Isomorphism

Composite of Endomorphisms is Endomorphism

Composite of Automorphisms is Automorphism

Definition:Canonical Epimorphism

Inversion Mapping is Automorphism iff Group is Abelian

Definition:Kernel of Group Homomorphism

Definition:Image of Subset under Mapping

Definition:Image Set of Mapping

Definition:Preimage of Subset under Mapping

Kernel is Trivial iff Monomorphism/Group

Definition:Closure (Abstract Algebra)/Algebraic Structure

Subgroups of Additive Group of Integers

Kernel of Group Homomorphism is Subgroup

Image of Group Homomorphism is Subgroup

Group Homomorphism Preserves Subgroups

Definition:Automorphism Group of Group

Automorphism Group is Subgroup of Symmetric Group

Intersection of Subgroups is Subgroup/General Result

Definition:Generator of Subgroup

Definition:Finitely Generated Algebraic Structure

Definition:Word (Abstract Algebra)

Integers under Addition form Infinite Cyclic Group