group

A group is an algebraic structure consisting of a set equipped with a binary operation such that the set is closed under the operation, the operation is associative, there is an identity element, and every element of the set has an inverse element.

Formally,
A group is a set $G$ equipped with a binary operations such that for all elements $a,b,c\in R$,

1. $G$ is closed: $ab\in G$
2. $G$ is associative: $(ab)c=a(bc)$
3. $G$ has an identity element: $ae=a=ea$
4. $G$ has inverses: $a(a^{-1})=e=(a^{-1})a$

A group is a monoid with inverses.
A group which is commutative is an abelian group.

The identity element of a group is unique.
The inverse of an element in a group is unique.
Group Cancellation Laws
Powers distribute over an operation if and only if the group is abelian.
If a group is finite, then every element has finite order.

Examples

Dihedral group $D_n$ : symmetries of $n$-gon
$\mathbb{R},\mathbb{Q},\mathbb{Z}$ with addition
$\mathbb{R}\setminus \{0\}$ with multiplication
$\mathbb{Z}/n\mathbb{Z}:=$ cyclic group ${\mathbb{Z}}_n$ with modular arithmetic
$n\times n$ matrices $M_n(\mathbb{R})$ with addition
General Linear group $G{L}_n(\text{any field }\mathbb{F})=\{A\in M_n(\mathbb{F}):det(A)\ne 0\}$