A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory is the study of groups.

A group consists of a base set $G$ and an operation $\ast : G\times G\to G$, such that

  1. $(a \ast b) \ast c = a \ast (b\ast c)$ for all $a,b,c\in G$ (associativity).
  2. There is an identity or unit element $e\in G$ with $e\ast a = a\ast e = a$ for all $a\in G.$
  3. For each element $a\in G$ there is an inverse element $a'$ such that $a\ast a' = a'\ast a = e$.

If additionally the commutative law $a \ast b = b\ast a$ for all $a,b\in G$ is satisfied, the group is called abelian or commutative.

The identity and inverses are always uniquely determined.

There are two main variants for the notation:

  1. In multiplicative notation, the operation is denoted by $a\cdot b$ or just $ab$, the identity is often denoted by $1$, and the inverse of an $a\in G$ is denoted by $a^{-1}$.
  2. For abelian groups often additive notation is used. Here, the operation is denoted by $a + b$, the identity by $0$ and the inverse of $a\in G$ by $-a$.

Group theory can also be seen as the mathematical theory of symmetries.

The historical roots of group theory include the study of symmetries of geometrical objects like the Platonic solids, and the study of roots of polynomial equations originated by √Čvariste Galois.