The Theory of Groups

The theory of groups is a branch of mathematics that studies structures composed of a set of elements and an operation (like addition or multiplication) that combines two elements to produce another. This structure must satisfy four key properties: closure (any operation stays within the set), associativity (grouping operations doesn’t change the result), an identity element (a neutral element that doesn’t change others), and inverses (every element has a partner that reverses its effect). Groups help us understand symmetry, patterns, and fundamental structures in mathematics and science, providing a formal way to analyze how objects and transformations relate.