# Lagrange's Theorem

**Lagrange's theorem** is a result on the indices of cosets of a group.

**Theorem.** Let be a group, a subgroup of , and a subgroup of . Then

*Proof.* For any , note that ; thus each left coset mod is a subset of a left coset mod ; since each element of is in some left coset mod , it follows that the left cosets mod are unions of left cosets mod . Furthermore, the mapping induces a bijection from the left cosets mod contained in an arbitrary -coset to those contained in an arbitrary -coset . Thus each -coset is a union of -cosets, and the cardinality of the set of -cosets contained in an -coset is independent of the choice of the -coset. The theorem then follows.

By letting be the trivial subgroup, we have In particular, if is a finite group of order and is a subgroup of of order , so the index and order of are divisors of .