## Proof of Lagrange's Theorem

Lagranges theorems says that the order of any subgroup of a finite group divides the order of (the order of a group is the number of distinct elements of ).

The idea of the proof is to take any subgroup and form all the subsets of  The subsets are called the left cosets of in (we could as well use the right cosets of in   ). Any two left cosets are identical or have no elements in common.

Suppose that is a subgroup of a group and is a left coset of If then for some Let be any element of then so that and so that Conversely, suppose that then no element of is in since if then a contradiction.

This means that all elements of are in or none are, ie the cosets are identical or disjoint.

Suppose that then The elements of are all distrinct, since if  which is a contradiction since all the elements of are distinct. The number of elemtns of each coset of is eaqual to the number of elements of Finally, the union of and all the cosets of is equal to since if then since The proof is now trivial since if there are elements of and elements of there are elemtns of each coset. Take the union of all the distinct cosets and add the orders to give so  