Groups, Rings, Integral Domains and Fields

Functions, mappings, maps, transformations, operators

Divisors, factors and multiples of integers; Common divisor, Greatest common divisor, Common multiple, Least common multiple, Division Algorithm, Euclid's algorithm, Unique Factorization Theorem

Congruence residue classes of integers modulo m

Products, quotients and roots of complex numbers in polar form, primitive roots of unity, cyclotomic polynomials

Modern, abstract mathematics

Binary operations

Groups, subgroups, complexes, cosets, transforms, normal subgroups, quotient groups, commutators, composition series, isomorphisms, homomorphisms, automorphisms

Permutations, permutation groups, transpositions, symmetric group

Cosets

Conjugates, conjugate classes, automorphisms, normal subgroups, quotient groups, homomorphic mapping of groups

The dihedral group of the square

Rings

Ideals, quotient rings, homomorphisms

Integral domains and fields

Axioms satisfied by rings, integral domains and fields

Isomorphisms, Automorphisms, Homomorphisms