Langrange's theorem (group theory)

Lagrange's theorem in group theory states that in a finite group, the number of elements (called the order) of any subgroup divides the total number of elements in the entire group. For example, if a group has 12 elements, then any subgroup within it must have 1, 2, 3, 4, 6, or 12 elements. This theorem helps mathematicians understand the structure of groups by linking the sizes of subgroups to the size of the entire group, providing a foundational insight into how larger groups are built from smaller, well-structured parts.