Group

Idea

A group is an algebraic structure used to abstract the concept of symmetry. It consists of a set equipped with a single binary operation that is associative, has an identity element, and provides an inverse for every element.

Definition

A group is a tuple $(G,\cdot ,e,{\_}^{-1})$ where $G$ is a set (or 0-type), $\cdot :G\times G\to G$ is a binary operation, $e:G$ is the identity element, and ${\_}^{-1}:G\to G$ is the inverse operation, satisfying: \forall x, y, z \in G. (x \cdot y) \cdot z = x \cdot (y \cdot z) \tag{assoc} \forall x \in G. e \cdot x = x \text{ and } x \cdot e = x \tag{ident} \forall x \in G. x^{-1} \cdot x = e \text{ and } x \cdot x^{-1} = e \tag{inverse}

Properties

• The identity element $e$ is unique.
• For every $x\in G$, the inverse $x^{-1}$ is unique.
• The inverse of a product follows the reversal rule: $(x\cdot y{)}^{-1}=y^{-1}\cdot x^{-1}$.
• Cancellation laws: $a\cdot x=b\cdot x⟹a=b$ and $x\cdot a=x\cdot b⟹a=b$.
• All groups are a one-element groupoid

Concepts

• Subgroup
• normal subgroup
• group homomorphism

References

• pinter2010-abstract-algebra
• riehl2016-category-theory