Lagrange's theorem, in the mathematics of grouping theory, states that for any finite group G, the arrange (number of elements) of all subgroup H of G divides the order of G. Lagrange's theorem is named after Joseph Lagrange.
Lagrange did not prove Lagrange's theorem in its general form. What he in fact proved was that if a polynomial in n variables has its variables permuted in all n! Ways, the number of various polynomials that are obtained is constantly a factor of n! The number of that kind of polynomials is the index in the symmetric group Sn of the subgroup H of permutations which preserve the polynomial. So the size of H divides n! With the later improvement of abstract groups, this end result of Lagrange on polynomials was known to extend to the general theorem about finite groups which currently bears his name.
Lagrange did not prove Lagrange's theorem in its general form. What he in fact proved was that if a polynomial in n variables has its variables permuted in all n! Ways, the number of various polynomials that are obtained is constantly a factor of n! The number of that kind of polynomials is the index in the symmetric group Sn of the subgroup H of permutations which preserve the polynomial. So the size of H divides n! With the later improvement of abstract groups, this end result of Lagrange on polynomials was known to extend to the general theorem about finite groups which currently bears his name.
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