---
tag: G5S1M9
kind: concept
name: Group
msc: [20A05]
sources:
  - book: isbn:9780471433347
    locator: Section 1.1
---

## Summary

A group is a set $G$ with a binary operation $\cdot$ that has these three properties.
The operation is associative: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c$ in $G$.
There is an identity element $e$ in $G$ such that $e \cdot a = a \cdot e = a$ for all $a$ in $G$.
Each element $a$ of $G$ has an inverse $a^{-1}$ in $G$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$.

The order of a finite group is its number of elements.

## Remarks

[[Lagrange's theorem|3C53TD]] gives a property of the order of a finite group.