# Group theory

Group theory is the study of groups, sets paired with an binary operation that satisfy the four group axioms.

# Basic Group Definition

**Groups** are algebraic structures consisting of a set, along with a binary operation (known as the group law of the set) that satisfy closure, associativity, identity, and invertibility (four group axioms). The group can be expressed as $(S, \cdot)$, where $S$ is the set and $\cdot$ is the operation. A basic example is the set of integers along with the addition operation:

- Closure: ensures the binary operation, when applied to any two elements from the set, produces a third element that is also part of the set. In the example, adding two integers always yields an integer.
- Associativity: $(a+b)+c = a+(b+c)$ for integers $a$, $b$, and $c$
- Identity: for any integer $a$, $a+0=a$. $0$ is the
*identity element*of addition. - Invertibility: for any integer $a$, there exists an integer $b$ such that $a+b=0$. $b$ is the
*inverse element*of $a$, i.e. $b = -a$.

Rings, fields, and vector spaces can be classified as groups with additional operations and axioms.

# Permutation Groups

For a given set $M$, the group $(G,\circ)$ is a permutation group acting on $M$. Here $G$ is a set of bijections from $M$ to itself (i.e. permutations), and the group operation is the composition of “permutation functions” in $G$. Additionally, if $G$ contains *all* possible permutations, then $(G,\circ)$ is a symmetric group (commonly written $\text{Sym}(M)$). All permutation groups on $M$ are *subgroups* of the symmetric group $\text{Sym}(M)$.

So permutation groups are a useful way to think about and represent transformations (say, to vertices of a polygon). For example (using an example from the Wikipedia article on permutation groups), let the vertices of a square be labeled 1,2,3,4 in counterclockwise fashion beginning at the top left vertex. Then the permutation (using cyclic notation) (1234) represents a 90 degree CCW rotation of the square (i.e. each vertex is mapped to the nearest CCW vertex, 1 -> 2, 2 -> 3, etc). The reflection of the square about the vertical center line is (14)(23).

# Finite Simple Groups

Groups can, in a sense, be broken down, or “factorized” (like integers can be broken down into prime factors). *Simple groups* are the basic building blocks of symmetric groups not unlike the way prime numbers are building blocks of any integer. Simple groups have normal subgroups including only the trivial group and itself. *The classification of finite simple groups* was a massive mathematical achievement uncovering a total of 18 families to which finite simple groups belong. There are also 26 exceptions, called the *sporadic groups*, defining finite simple groups that do not belong to any of 18 families mentioned above.

# Monster group

By far the largest of the sporadic groups, a fascinating yet mysterious symmetric group of a 196883-dimensional object. See the great video from Numberphile and John Conway (along with a new one from Grant)

# Lie Groups

# Abelian Groups

Abelian groups, or commutative groups, are groups where the order of two group elements does not matter when applying the group operation (i.e. is commutative).