Group

Overview

A group records operations that can be composed and undone. Its elements might be integers, invertible matrices, or symmetries of a figure. The operation belongs to the data: the integers under addition form a group, whereas the same set under ordinary multiplication does not. Understanding that distinction prevents treating a familiar set as an algebraic structure before specifying its operation.

Prerequisites

Use sets, functions, and composition of functions. A binary operation on $G$ is a function $G\times G\to G$, so closure is already built into this notation. A monoid has an associative operation and an identity; a group adds invertibility of every element.

Definition and conventions

A group consists of a set $G$, an operation written $ab$, and an element $e$ such that, for all $a,b,c\in G$,

and each $a$ has an element $a^{-1}$ satisfying $aa^{-1}=a^{-1}a=e$. These axioms force the identity and each inverse to be unique. Commutativity is an additional condition, $ab=ba$; a group satisfying it is abelian. Additive notation uses $0$ and $-a$ in place of $e$ and $a^{-1}$. For permutations, this article composes from right to left.

Intuition

Think of an element as a reversible instruction. Associativity says that batching consecutive instructions does not change the result. It says nothing about exchanging their order. Reversibility is the decisive difference between a group and a general collection of operations: a projection loses information, but a rotation can be undone.

Examples and non-examples

In $(\mathbb Z,+)$, the inverse of $7$ is $-7$. In $(\mathbb Z/5\mathbb Z,+)$, the inverse of the class of $2$ is the class of $3$, since $2+3\equiv0\pmod5$.

The permutations of three letters give a nonabelian example. With the convention above, $(12)(23)$ sends $1$ to $2$, whereas $(23)(12)$ sends $1$ to $3$. Thus these operations do not commute. Nonnegative integers under addition form a monoid but not a group: $1$ has no additive inverse within that set. All square matrices under multiplication also fail the inverse axiom because the zero matrix is singular.

Important properties

Cancellation follows from inverses. If $ab=ac$, multiplying on the left by $a^{-1}$ gives $b=c$. Likewise right cancellation holds. The inverse of a product reverses the order:

Indeed, multiplying either way gives the identity. The reversal matters in nonabelian groups. For finite groups, cosets of a subgroup partition the group into equal-sized pieces; consequently the subgroup's order divides the group's order. Finiteness is essential to that numerical statement.

Connections and further reading

Group actions turn this abstract operation into actual transformations of sets. Normal subgroups permit quotient groups, and representations turn group elements into invertible linear maps. Read Milne, Chapter 1 for the foundational definitions and permutation examples before proceeding to these constructions.