Lagrange's theorem (group theory)
|Algebraic structure → Group theory|
Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.
This can be shown using the concept of left cosets of H in G. The left cosets are the equivalence classes of a certain equivalence relation on G and therefore form a partition of G. Specifically, x and y in G are related if and only if there exists h in H such that x = yh. If we can show that all cosets of H have the same number of elements, then each coset of H has precisely |H| elements. We are then done since the order of H times the number of cosets is equal to the number of elements in G, thereby proving that the order of H divides the order of G.
To show any two left cosets have the same cardinality, it suffices to demonstrate a bijection between them. Suppose aH and bH are two left cosets of H. Then define a map f : aH → bH by setting f(x) = ba−1x. This map is bijective because it has an inverse given by
This proof also shows that the quotient of the orders |G| / |H| is equal to the index [G : H] (the number of left cosets of H in G). If we allow G and H to be infinite, and write this statement as
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
Lagrange's theorem can also be used to show that there are infinitely many primes: if there were a largest prime p, then a prime divisor q of the Mersenne number would be such that the order of 2 in the multiplicative group (see modular arithmetic) divides the order of , which is . Hence , contradicting the assumption that p is the largest prime.
Existence of subgroups of given order
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.
A "Converse of Lagrange's Theorem" (CLT) group is a finite group with the property that for every divisor of the order of the group, there is a subgroup of that order. It is known that a CLT group must be solvable and that every supersolvable group is a CLT group. However, there exist solvable groups that are not CLT (for example, A4) and CLT groups that are not supersolvable (for example, S4, the symmetric group of degree 4).
There are partial converses to Lagrange's theorem. 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).
Disproving the converse of Lagrange's theorem
The converse of Lagrange's theorem states that if is a divisor of the order of a group , then there exists a subgroup where .
so the divisors are . Assume to the contrary that there exists a subgroup in with .
Let . Since both and are subgroups of , is also a subgroup of .
From Lagrange's theorem, the order of must divide both and , the orders of and respectively. The only two positive integers that divide both and are and . So or .
Assume , then . If does not share any elements with , then the 5 elements in besides the Identity element must be of the form where are distinct elements in .
Since any element of the form squared is , and , any element of in the form must be paired with its inverse. Specifically, the remaining 5 elements of must come from distinct pairs of elements in that are not in . This is impossible since pairs of elements must be even and cannot total up to 5 elements. Thus, the assumptions that is wrong, so .
Then, where , must be in the form where are distinct elements of . The other four elements in are cycles of length 3.
Note that the cosets generated by a subgroup of a group is a partition of the group. The cosets generated by a specific subgroup are either identical to each other or disjoint. The index of a subgroup in a group is the number of cosets generated by that subgroup. Since and , will generate two left cosets, one that is equal to and another, , that is of length 6 and includes all the elements in not in .
Since there are only 2 distinct cosets generated by , then must be normal. Because of that, . In particular, this is true for . Since .
Without loss of generality, assume that . Then . Transforming back, we get . Because contains all disjoint transpositions in , . Hence, .
Since , we have demonstrated that there is a third element in . But earlier we showed that , 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 and the converse of Lagrange's theorem is not necessarily true.
Lagrange did not prove Lagrange's theorem in its general form. He stated, in his article Réflexions sur la résolution algébrique des équations, 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!. (For example, if the variables x, y, and z are permuted in all 6 possible ways in the polynomial x + y − z then we get a total of 3 different polynomials: x + y − z, x + z − y, and y + z − x. Note that 3 is a factor of 6.) The number of such polynomials is the index in the symmetric group Sn of the subgroup H of permutations that preserve the polynomial. (For the example of x + y − z, the subgroup H in S3 contains the identity and the transposition (x y).) So the size of H divides n!. 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. In 1844, Augustin-Louis Cauchy proved Lagrange's theorem for the symmetric group Sn.
- Martin Aigner and Günter M. Ziegler: Proofs from THE BOOK, Springer, Berlin, 1998, Chapter 1
- Lagrange, Joseph-Louis (1771). "Suite des réflexions sur la résolution algébrique des équations. Section troisieme. De la résolution des équations du cinquieme degré & des degrés ultérieurs" [Series of reflections on the algebraic solution of equations. Third section. On the solution of equations of the fifth degree & higher degrees]. Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin: 138–254. ; see especially pages 202-203.
- Gauss, Carl Friedrich (1801), Disquisitiones Arithmeticae (in Latin), Leipzig (Lipsia): G. Fleischer, pp. 41-45, Art. 45-49.
- Augustin-Louis Cauchy, §VI. — Sur les dérivées d'une ou de plusieurs substitutions, et sur les systèmes de substitutions conjuguées [On the products of one or several permutations, and on systems of conjugate permutations] of: "Mémoire sur les arrangements que l'on peut former avec des lettres données, et sur les permutations ou substitutions à l'aide desquelles on passe d'un arrangement à un autre" [Memoir on the arrangements that one can form with given letters, and on the permutations or substitutions by means of which one passes from one arrangement to another] in: Exercises d'analyse et de physique mathématique [Exercises in analysis and mathematical physics], vol. 3 (Paris, France: Bachelier, 1844), pp. 183-185.
- Jordan, Camille (1861). "Mémoire sur le numbre des valeurs des fonctions" [Memoir on the number of values of functions]. Journal de l'École Polytechnique. 22: 113–194. Jordan's generalization of Lagrange's theorem appears on page 166.
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- Gallian, Joseph (2006), Contemporary Abstract Algebra (6th ed.), Boston: Houghton Mifflin, ISBN 978-0-618-51471-7
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