In mathematics, **Lagrange's theorem** states that if *G* is a finite group and *H* is a subgroup of *G*, then the order (that is, the number of elements) of *H* divides the order of *G*.

This can be shown using the concept of left cosets of *H* in *G*. The left cosets are the equivalence classes of a certain equivalence relation on *G* and therefore form a partition of *G*. If we can show that all cosets of *H* have the same number of elements, then we are done, since *H* itself is a coset of *H*. Now, if *aH* and *bH* are two left cosets of *H*, we can define a map *f* : *aH* → *bH* by setting *f*(*x*) = *ba ^{-1}x*. This map is bijective because its inverse is given by

This proof also shows that the quotient of the orders |*G*| / |*H*| is equal to the index **[***G***:***H***]** (the number of left cosets of *H* in *G*). If we write this statement as

- |
*G*| =**[***G***:***H***]**· |*H*|,

A consequence of the theorem is that the order of any element *a* of a finite group (i.e. the smallest positive integer *k* with *a*^{k} = *e*) divides the order of that group, since the order of *a* is equal to the order of the subgroup generated by *a*. If the group has *n* elements, it follows

*a*^{n}=*e*.