$group

A $group in mathematics is a set of elements combined with an operation (like addition or multiplication) that satisfies four key properties: there's an identity element that leaves others unchanged, every element has an inverse that reverses its effect, the operation is associative (grouping doesn't matter), and the set is closed under the operation (combining any two elements results in another within the set). Groups provide a way to study symmetry and structure in various mathematical contexts, from numbers to geometric patterns, making them fundamental to understanding many areas of math and science.