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See also: Theorem of Lagrange

In Theory of the groups, the theorem of Lagrange indicates an elementary result providing of information Combinatoire S on the finished groups: the cardinal of a sub-group of a finished group divides the cardinal of the group.

Theorem of Lagrange: For a G group finished, and any sub-group H of G, the cardinal (order) of H divides the cardinal of G:

\ mbox {card} (H) \ mid \ mbox {card} (G) .

The quotient of the cardinal of G by the cardinal of H is called the index H in G , it is noted:

\ mbox {card} (G) = \ mbox {card} (H) \ times \, .

Demonstrations

The first Démonstration is based on the concept of action of group whose definition is pointed out. The second does not require little vocabulary concerning the theory of the groups.

Action of group

See also: Action of group

An action of a group finished G on a unit X is the data for any element G of G of a bijective mapping X \ rightarrow X sending X on an element noted g.x, checking the following condition:

\ forall G, H \ in G, \; G. (h.x) = (gh) .x
If the group finished G acts on a unit X , the stabilizer of a point X of X is defined like the whole of the elements H of the group G such as h.x=x; it is easy to check that the stabilizers are sub-groups of G . The orbit of X is defined as the whole of the elements there of X being written: g.x=y for g \ in G. The conjugation by the element G induces a bijection of the stabilizer of X on the stabilizer of g.x:
Stab (g.x) =g.Stab (X) .g^ {- 1} = \ {ghg^ {- 1}, H \ in Stab (X) \} .
In particular, the cardinal of the stabilizer of X does not depend on the choice of X in his orbit. By application of the Lemma of the shepherds, the cardinal of the stabilizer of X divides the cardinal of G , and the quotient is equal to the cardinal of the orbit of X .

The proof of the theorem of Lagrange consists in carrying out any sub-group H of a group finished G like a stabilizer for an action of G . The action considered is the action of the group G by translation on the left on the unit X of its parts, definite as follows:

\ forall G \ in G, \ forall has \ in X, \; g.A= \ {ga, has \ in has \} .
The stabilizer of a sub-group H is présisément H . Indeed, the equality gH=H implies the existence of an element H checking gh=1; in other words, g must be the reverse of an element of H , and a fortiori must belong him even to H . Reciprocally, like the product of elements of H belongs to H , any element H of H checks h.H=H. In fact, by double inclusion, H is the stabilizer of H and the property is some shown.

Relation of equivalence

The second proof of the theorem consists with partitionner the G unit of a family of equipotent whole to H (in other words, of the same cardinal than H ). The data of a partition is equivalent to the data of a Relation of equivalence on G.

That is to say \ mathcal R the binary relation on G defined by:

x \ mathcal R there \ Leftrightarrow xy^ {- 1} \ in H
\ mathcal R checks the following properties:
  • Reflexivity : By definition of the Opposite , any element x in G checks the identity x.x^ {- 1} =e. The neutral element e belongs to H by definition of a sub-group. In fact, x \ mathcal Rx.
  • Symmetry : For all x and y in G, one writes:
    x \ cdot y^ {- 1} = \ bigl (there \ cdot x^ {- 1} \ bigr) ^ {- 1} .
    Of continuation, x.y^ {- 1} belongs to H if and only if y.x^ {- 1} belongs to H; equivalence which is rewritten: xRy\Leftrightarrow yRx.
  • Transitivity : For all x, there, z in G, the associativeness of the product gives:
    x \ cdot z^ {- 1} = \ bigl (X \ cdot y^ {- 1} \ bigr) \ cdot \ bigl (there \ cdot z^ {- 1} \ bigr) .
    the H part being stable by product, if x \ mathcal R y and y \ mathcal R z, then x \ mathcal R z.
The relation \ mathcal R is reflexive, symmetrical, and transitive, therefore it defines a relation of equivalence on the G unit. The class of equivalence of e is not other than H.

Let us notice that the relation \ mathcal R is G-invariante on the right: for all x, y and z in G, x \ mathcal R y imply (thus, is equivalent to) (xz) \ mathcal R (yz) . In particular, when C indicates a class of equivalence, and x an element of C, then the unit C.x^ {- 1} is the class of equivalence of e, therefore H. As the application y \ mapsto yx^ {- 1} is a bijection, the units C and H has even cardinal. Of continuation, the classes of equivalence partitionnent G in parts in the same way cardinal than H.

History

The French mathematician Joseph-Louis Lagrange (January 25th 1736 - April 10th 1813) showed that, by permutation variable N of a polynomial expression, the number of expressions obtained is a divider of n! . The whole of the permutations is seen today like a group with n! elements, acting on the polynomials with variable N . The work of Lagrange is reinterpreted like the calculation of the cardinal of an orbit of this action: it seems precursor of research on the groups, whose first terms of vocabularies were introduced following work of Augustin Louis Cauchy (August 21st 1789 - May 23rd 1857) , and whose formal definition was given only in 1882 by Walther Franz Anton von Dyck (December 6th 1856 - November 5th 1934) .

Current applications

  • the order of an element X of a finished group is defined as the cardinal of the sub-group which it generates. It is the smallest natural entirety checking N : x^n=e. It divides the order of the group.

  • a group G of order first p is cyclic. Indeed, any element not no one X of G is of a nature strictly higher than 1 and by what precedes a divider by p . As p is first, the order of X is p ; in other words, X generates a cyclic group of order p , necessarily equal to G .

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