Lagrange's theorem (group theory)

In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group G, the order (number of elements) of every subgroup of G divides the order of G. 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 . Lagrange's theorem This variant holds even if is infinite, provided that , , and are interpreted as cardinal numbers. The left cosets of H in G are the equivalence classes of a certain equivalence relation on G: specifically, call x and y in G equivalent if there exists h in H such that x = yh. Therefore, the left cosets form a partition of G. Each left coset aH has the same cardinality as H because defines a bijection (the inverse is ). The number of left cosets is the index [G : H]. By the previous three sentences, Lagrange's theorem can be extended to the equation of indexes between three subgroups of G. Extension of Lagrange's theorem Let S be a set of coset representatives for K in H, so (disjoint union), and . For any , left-multiplication-by-a is a bijection , so . Thus each left coset of H decomposes into left cosets of K. Since G decomposes into left cosets of H, each of which decomposes into left cosets of K, the total number of left cosets of K in G is . If we take K = (e is the identity element of G), then [G : ] = G and [H : ] = H. Therefore, we can recover the original equation 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 number k with ak = e, where e is the identity element of the group) divides the order of that group, since the order of a is equal to the order of the cyclic subgroup generated by a. If the group has n 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.