Group theory

In mathematics, a group is a nonempty set $\mathsf{G}$ equipped with a binary operation $(g_1,g_2)\mapsto g_1\cdot g_2$ that combines two elements of the group to form a third element also belonging to the group.

Groups have the following key properties:

• Closure; $a\cdot b\in \mathsf{G};\forall a,b\in G$
• Associativity; $a\cdot (b\cdot c)=(a\cdot b)\cdot c,\forall a,b,c\in \mathsf{G}$
• Identity element; $\exists e\in \mathsf{G}\ni g\cdot e=e\cdot g=g\forall g\in \mathsf{G}$
• Inverse element; for each $g\in \mathsf{G}$, there exists $g^{-1}\in \mathsf{G}\ni g\cdot g^{-1}=g^{-1}\cdot g=e$

A group is Abelian if $a\cdot b=b\cdot a\forall a,b\in \mathsf{G}$.

Terminology

A binary operation on a set $S$ is a map $B:S\times S\to S$. A pair $(S,B)$ where $B$ is a binary operation on $S$ is a magma.

Some structures that are more basic than a group:

• Semigroups are nonempty sets with a binary operation on the set with the associativity property.
• Monoids are semigroups with an additional identity element.

i.e., semigroups are subsets of monoids are subsets of groups.