Group (Mathematics)

up:: 021 MOC Algebra

A group is a triple $(G,\cdot )$, consisting of a set $G$ imbued with an operation $\cdot :G\times G\to G$, in such a way that

1. Closed under operation: $\forall g,h\in G,g\cdot h\in G$
2. Existence of neutral element: $\exists e\in G\mid \forall g\in G,g\cdot e=e\cdot g=g$
3. Existence of inverse element: $\forall g\in G,\exists g^{-1}\in G\mid g\cdot g^{-1}=g^{-1}\cdot g=e$

Properties

• A group’s identity is unique
• All elements of a group have a unique inverse

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References

• Group (mathematics) - Wikipedia