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Section Definitions and Algorithms

This page contains definitions of important terms and algorithms used throughout the course.

Subsection Definitions

Definition 4 Symmetry
Definition 10 Binary Operation
Definition 11 Abelian group
Definition 15 Subgroup
Definition 20 Center of a Group
Definition 23 Order of a Element
Definition 25 Cyclic Group
Definition 30 Cyclic Subgroup
Definition 34 Permutation
Definition 35 Permutation Group
Definition 36 Symmetric Group
Definition 44 Cycle Type of a Permutation
Definition 48 Group Isomorphism
Definition 49 Alternating Group
Definition 57 Automorphism of a Group
Definition 59 Cosets of a Subgroup
Definition 61 Index of a Subgroup
Definition 64 Inner Automorphism
Definition 76 External Direct Product of Groups
Definition 87 Normal Subgroup
Definition 91 Quotient Group
Definition 97 Group Homomorphism
Definition 98 Kernel of a Homomorphism
Definition 116 Group Action
Definition 117 Group Action Features
Definition 123 Centralizer and Normalizer
Definition 127 Sylow Subgroups
Definition 140 Zero-Divisor
Definition 141 Integral Domain
Definition 145 Division in a Ring
Definition 148 Characteristic of a Ring
Definition 156 Principal Ideal
Definition 157 Prime and Maximal Ideals
Definition 162 Ring Homomorphism
Definition 171 Ring of Polynomials over \(R\)
Definition 172 Addition and Multiplication in \(R[x]\)
Definition 173 Polynomial Properties
Definition 180 Irreducible and Reducible Polynomials
Definition 182 Primitive Polynomials
Definition 194 Irreducibles and Primes in Domains
Definition 198 Unique Factorization Domain
Definition 203 Euclidean Domain

Subsection Algorithms