Lagrange's theorem (group theory)

Lagrange's theorem in group theory states if G is a finite group and H is a subgroup of G, then |H| (how many elements are in H, called the order of H) divides |G|. Moreover, the number of distinct left (right) cosets of H in G is |G|/|H|. This theorem is named after mathematician Joseph-Louis Lagrange.

Applications

• For any g in a group G, $g^{k}=e$ for some k that divides the |G|
• Any group of prime order cyclic (Any element in G can be created by a single element) and simple (no normal subgroups that aren't trivial)