Introduction to Group Theory

An algebraic structure is a set with a collection of operations (with finite numbers of inputs) that may act upon it.
Groups are a particular subset of algebraic structures: they must have at least one binary operator and satisfy the following four axioms / requirements:
  1. Associativity: `\forall a,b,c\in\mathbb X, (a \oplus b) \oplus c = a \oplus (b \oplus c)`
  2. Closure: `\forall a, b \in \mathbb X, a \oplus b = c \implies c\in\mathbb X`
  3. Identity Element: `\exists i\in\mathbb X: \forall a\in\mathbb X, a\oplus i = i\oplus a = a`
  4. Inversion: `\forall a\in\mathbb X, \exists b\in\mathbb X: a\oplus b = b\oplus a = i`, where `i` is the identity element
(On this page, `\mathbb X` represents an arbitrary set and `\oplus` an arbitrary operator)
TODO: Link set notation

Groups can be defined using set notation to describe the set and the operator, and this combination forms group theory notation. [src]
For instance, `(\mathbb Z, +)` denotes the "set of integers under addition" and `(\mathbb Q \setminus\{0\}, \cdot)` notates the "set of rationals (excluding zero) under multiplication".
In general, this can be written as `(\mathbb X, \oplus)`.
Important Note: `(\mathbb Q, \cdot)` is not a group because there is no element `b` for which `0\cdot b=1`. [src]

Rings are composed of a set and two binary operations, and they obey the "ring axioms": [src]
  1. The set is an Abelian group under addition.
  2. Multiplication is associative and there exists a multiplicative identity.
  3. Multiplication is distributive over addition.
Multiplication must maintain closure, but it is not necessarily commutative and does not need to be invertible.
A ring without a multiplicative identity is called a rng, or alternatively, a pseudo-ring ("pseudo-ring" is slightly ambiguous and should be avoided). [src]
A ring in which the operation of multiplication is commutative is said to be a commutative ring.
In a commutative ring, if there is an element `a` such that `a\neq 0` and `\exists b:a\cdot b=0`, `a` is said to be a zero divisor.
A commutative ring with no zero divisors is known as an integral domain, and an example is the set of integers `\mathbb Z`. [src]

Fields are rings in which the elements of a ring (excluding the zero) comprise an Abelian group with the second operation.
This means that in addition to the ring axioms, fields have commutativity of multiplication and multiplicative inverses.
An example of a field is the set of rational numbers, `\mathbb Q`.

Important Note: In the context of group theory, `0` and `1` are often defined as the additive and multiplicative identities, where "addition" [`+`] and "multiplication" [ `\cdot` ] are default names and symbols given to the two operations involved. These often, but not always, coincide with the conventional usages.

Key Words

Abelian groups are those for which the operator is commutative: `\forall a,b\in\mathbb X, a\oplus b=b\oplus a`.
Subgroups are groups that are subsets of larger groups, still obeying all four key axioms.
Group homomorphisms are functions `f:\mathbb X\to\mathbb Y` such that, with groups `(\mathbb X, \oplus)` and `(\mathbb Y, \boxdot)` and `a,b\in\mathbb X, f(a\oplus b)=f(a)\boxdot f(b)`.
Isomorphisms are a subset of homomorphisms, those which have inverse functions that are also homomorphisms.
This implies that isomorphisms are bijective (homo)morphisms. Composing isomorphisms forms an "identity function".
Two groups that admit an isomorphism are called "isomorphic". This is written as `\mathbb A \cong\mathbb B` for groups with underlying sets `\mathbb A` and `\mathbb B`. [src]

Automorphisms are isomorphisms of a object, often a group, which map the object unto itself. [src]
The set of these (along with function composition) forms an "automorphism group". [src]
Important Note: This will form a group, no matter if the original object was a group, ring, module, or other structure. [src]
Automorphism groups must satisfy the following four requirements, analogous to those identified for groups above:
  1. Associativity: Composition of automorphisms is associative.
  2. Closure: Composing any two automorphisms will produce another automorphism.
  3. Identity Element: The identity function is an automorphism, so it is included by definition.
  4. Inversion: All automorphisms in the group have inverse functions by definition.

TODO: Fully utilize http://mathworld.wolfram.com/Morphism.html, monoids and others from https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node6.html
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