Finite group theory is a branch of mathematics that studies groups with a finite number of elements. A group is a set equipped with an operation that combines any two elements to form a third element, satisfying certain conditions like closure, associativity, identity, and invertibility. Finite groups are important in various fields, including algebra, geometry, and number theory. One of the key results in finite group theory is Lagrange's theorem, which states that the order of a subgroup divides the order of the entire group. This theory also explores the classification of finite simple groups, which are the building blocks of all finite groups, similar to prime numbers in arithmetic.