Proof of Lagrange's Theorem

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

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

Suppose thatis a subgroup of a groupandis a left coset ofIfthenfor someLetbe any element ofthenso thatandso thatConversely, suppose thatthen no element ofis insince ifthena contradiction.

This means that all elements ofare inor none are, ie the cosets are identical or disjoint.

Suppose thatthenThe elements ofare all distrinct, since ifwhich is a contradiction since all the elements ofare distinct. The number of elemtns of each coset ofis eaqual to the number of elements of

Finally, the union ofand all the cosets ofis equal tosince if then since

The proof is now trivial since if there areelements ofandelements ofthere areelemtns of each coset. Take the union of all thedistinct cosets and add the orders to giveso