All my learning about abstract algebra, ring theory, field theory, lattice theory, module theory, category theory, Galois theory, homology, homotopy, univalent foundations, HoTT and K theory.
groupoid - a set with some operations (usually inverse) and closed under itself
semigroup - an associative groupoid
monoid - a semigroup with an identity
group - a monoid with inverses
abelian group - a commutative group
coset - a modification of a group/subgroup (not always forms a group)
left-coset - a coset with a left-side operation for non-abelian groups
right-coset - a coset with a right-side operation for non-abelian groups
normal group - a subgroup such that all left cosets are identicals to the same right coset
trivial group - the only 2 immutable subgroups of a group -- itself and its identity
nontrivial subgroup - the same as subgroups, except itself and its identity
quotient/factor group - the set of all cosets
congruence class - a quotient group which you add two elements of distinct sets and gets another set
cyclic group - a group that has a generator (i.e, it's generated by a single element)
homomorphism - convert a group into another group
isomorphism - a bijective homomorphism
kernel - a function between homomorphisms that collects all the elements in the 1st set that results in identity on the 2nd set
order of element - is denoted by |x| = n, where x^n = 0identity)
order of group - the number of elements that is contained within the group
symmetric/permutation group - a group Sn where n is the order of the group and can be rearranged in n! ways, and has a function that allows you to choose an element by its position
dihedral group - a permutation group of regular polygons where the notation D^n is the number of sides and D^2n is the number of symmetries and is cyclic under rotations and flips
simple group - a group with only two subgroups -- itself and its identity
ring - an abelian group with multiplication
zero divisor - some multiplicative operation that is out of range and points to identity
unit ring - a ring with some 1-like behaviour for multiplication
integral domain - a unit ring with no zero divisors
matrix group - a group of matrices over a field with scalar multiplication
ideal - a subset that still is closed under itself. e.g: set of even numbers, since all addition, subtraction and multiplication of even numbers results in another even number
vector space - the set of vectors
field - a ring with a multiplication inverse (division)
module - a generalized vector space, that has a ring of scalars (and therefore it no longer reqyures multiplication inverse)
Lagrange's theorem - says that the possibilities of subgroups is the number of all the divisors of the number of elements of the main group
symmetric groups of triangles (AKA dihedral group of order 6)