在数学中，群是一种代数结构，由一个集合以及一个二元运算所组成。

一个群必须满足一些被称为“群公理”的条件，也就是封闭性、结合律、单比特和逆元。很多熟知的数学结构比如数系统都遵从这些公理，例如整数配备上加法运算就形成一个群。如果将群公理的公式从具体的群和其运算中抽象出来，就使得人们可以用灵活的方式来处理起源于抽象代数或其他许多数学分支的实体，而同时保留对象的本质结构性质。

群在数学内外各个领域中是无处不在的，这使得它们成为当代数学的组成的中心原理。

群与对称概念共有基础根源。对称群把几何物体的如此描述物体的对称特征：它是保持物体不变的变换的集合。这种对称群，特别是连续李群，在很多学术学科中扮演重要角色。例如，矩阵群可以用来理解在狭义相对论底层的基本物理定律和在分子化学中的对称现象。

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely. It allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.