Lagrange's theorem (group theory)

In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then

, i.e. the order (number of elements) of every subgroup H divides the order of group G. The theorem is named after Joseph-Louis Lagrange.

, defined as the number of left cosets of

Lagrange's theorem — If H is a subgroup of a group G, then

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.[1] Extension of Lagrange's theorem — If H is a subgroup of G and K is a subgroup of H, then Let S be a set of coset representatives for K in H, so

left cosets of K, the total number

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.

These special cases were known long before the general theorem was proved.

The theorem also shows that any group of prime order is cyclic and simple, since the subgroup generated by any non-identity element must be the whole group itself.

[2] Lagrange's theorem raises the converse question as to whether every divisor of the order of a group is the order of some subgroup.

This does not hold in general: given a finite group G and a divisor d of |G|, there does not necessarily exist a subgroup of G with order d. The smallest example is A4 (the alternating group of degree 4), which has 12 elements but no subgroup of order 6.

For general groups, Cauchy's theorem guarantees the existence of an element, and hence of a cyclic subgroup, of order any prime dividing the group order.

Sylow's theorem extends this to the existence of a subgroup of order equal to the maximal power of any prime dividing the group order.

For solvable groups, Hall's theorems assert the existence of a subgroup of order equal to any unitary divisor of the group order (that is, a divisor coprime to its cofactor).

The converse of Lagrange's theorem states that if d is a divisor of the order of a group G, then there exists a subgroup H where |H| = d. We will examine the alternating group A4, the set of even permutations as the subgroup of the Symmetric group S4.

Let V be the non-cyclic subgroup of A4 called the Klein four-group.

The cosets generated by a specific subgroup are either identical to each other or disjoint.

Since |A4| = 12 and |H| = 6, H will generate two left cosets, one that is equal to H and another, gH, that is of length 6 and includes all the elements in A4 not in H. Since there are only 2 distinct cosets generated by H, then H must be normal.

Because V contains all disjoint transpositions in A4, gvg−1 ∈ V. Hence, gvg−1 ∈ H ⋂ V = K. Since gvg−1 ≠ v, we have demonstrated that there is a third element in K. But earlier we assumed that |K| = 2, so we have a contradiction.

Therefore, our original assumption that there is a subgroup of order 6 is not true and consequently there is no subgroup of order 6 in A4 and the converse of Lagrange's theorem is not necessarily true.

Lagrange himself did not prove the theorem in its general form.

He stated, in his article Réflexions sur la résolution algébrique des équations,[3] that if a polynomial in n variables has its variables permuted in all n!

ways, the number of different polynomials that are obtained is always a factor of n!.

The number of such polynomials is the index in the symmetric group Sn of the subgroup H of permutations that preserve the polynomial.

With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name.

In his Disquisitiones Arithmeticae in 1801, Carl Friedrich Gauss proved Lagrange's theorem for the special case of

, the multiplicative group of nonzero integers modulo p, where p is a prime.

[4] In 1844, Augustin-Louis Cauchy proved Lagrange's theorem for the symmetric group Sn.

[5] Camille Jordan finally proved Lagrange's theorem for the case of any permutation group in 1861.