Lagrange's theorem
Lagrange's theorem is a clean divisibility law for finite groups: the size of any subgroup must divide the size of the whole group evenly, leaving no remainder. If a group has 12 elements, every subgroup must have 1, 2, 3, 4, 6, or 12 elements — a subgroup of size 5 is simply impossible. It is one of the first surprising, powerful facts a beginner meets in group theory.
Formally, if G is a finite group and H is a subgroup, then the order of H divides the order of G. The proof works by chopping G into equal-sized translated copies of H called cosets: these copies are disjoint and identical in size, and together they exactly fill G, so the count of H must split the count of G.
The consequences cascade. Because every element generates a cyclic subgroup whose size is the element's order, that order must divide the group's order too. From there one quickly proves Fermat's little theorem in number theory. A warning about the converse: a divisor of the group's order need not correspond to an actual subgroup — Lagrange constrains, but does not guarantee existence.
The symmetric group S₃ has order 6. Its possible subgroup sizes are exactly the divisors of 6: 1, 2, 3, 6 — and indeed S₃ has subgroups of each of these sizes, but never one of size 4 or 5.
Subgroup orders of S₃ are limited to divisors of 6.
The converse is false in general: having a divisor d of the group's order does not guarantee a subgroup of order d exists. The alternating group A₄ has 12 elements but no subgroup of order 6, a famous counterexample. (Partial converses, like the Sylow and Cauchy theorems, do hold under extra conditions.)