In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group , the order (number of elements) of every subgroup of divides the order of . The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup of a finite group , not only is an integer, but its value is the index , defined as the number of left cosets of in . This variant holds even if is infinite, provided that , , and are interpreted as cardinal numbers. Proof The left cosets of in are the equivalence classes of a certain equivalence relation on : specifically, call and in equivalent if there exists in such that . Therefore, the left cosets form a partition of . Each left coset has the same cardinality as because defines a bijection (the inverse is ). The number of left cosets is the index . By the previous three sentences, Extension Lagrange's theorem can be extended to the equation of indexes between three subgroups of . If we take ( is the identity element of ), then and . Therefore, we can recover the original equation . Applications A consequence of the theorem is that the order of any element of a finite group (i.e. the smallest positive integer number with , where is the identity element of the group) divides the order of that group, since the order of is equal to the order of the cyclic subgroup generated by . If the group has elements, it follows This can be used to prove Fermat's little theorem and its generalization, Euler's theorem. These special cases were known long before the general theorem was proved. The theorem also shows that any group of prime order is cyclic and simple, since the subgroup generated by any non-identity element must be the whole group itself. Lagrange's theorem can also be used to show that there are infinitely many primes: if there were a largest prime , then a prime divisor of the Mersenne number would be such that the order of in the multiplicative group (see modular arithmetic) divides the order of , which is . Hence , contradicting the assumption that is the largest prime. Existence of subgroups of given order Lagrange's theorem raises the converse question as to whether every divisor of the order of a group is the