MathGloss

A group is a set $G$ together with a binary operation $\cdot:G\times G \to G$ satisfying the following properties:

1. $\cdot$ is associative;
2. there exists an identity element $e \in G$ such that $g\cdot e = g = e\cdot g$ for all $g \in G$.
3. for each $g \in G$ there exists an inverse element $g^{-1}$ such that $g\cdot g^{-1} = e = g^{-1}\cdot g$.

Groups can be thought of as

• associative magmas
• semigroups with an identity and with inverses
• monoids with inverses
• categories with a single object and every morphism an isomorphism

Wikidata ID: Q83478

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