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

  1. $(a \ast b) \ast c = a \ast (b\ast c)$ for all $a,b,c\in G$ (associative law)
  2. There is a neutral 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 neutral element and the inverse elements are always uniquely determined.

There are two main variants for the notation:

  1. In mutliplicative notation, the operation is denoted by $a\cdot b$ or just $ab$, the neutral element is often denoted by $1$, and the inverse element 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 neutral element by $0$ and the inverse element 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 Evariste Galois.

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