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Theorem

Lagrange's Theorem (Group Theory)

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Algebra
Circle Divided Into Equal Wedges

Lagrange's theorem states that if H is a subgroup of any finite group G, then the order of H is a divisor of the order of G, that is, the order of every subgroup divides the order of the whole group. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Disputed
Proof Year
1861 1
Lagrange himself stated only a special case, about permutations of polynomial variables, in his 1771 paper. The theorem was generalized to all groups later, through the work of Gauss in 1801 and Cauchy in 1844, and finally proved for all permutation groups by Camille Jordan in 1861.
Statement
For any finite group and any subgroup of that group, the number of elements in the subgroup always divides evenly into the number of elements in the whole group. 1
Classification
Statement Form
Characterization Theorem 1
Connections

Associated With

Group (Abstract Algebra), Concepts

The theorem relates the order of a subgroup to the order of its whole group, so its subject is the group concept itself.

Additional Source Lagrange's Theorem, Group Theory (Wikipedia)opening sentence
Sylow Theorems, Theorems

In Branch

Algebra, Branches of Mathematics
Source Lagrange's Theorem, Group Theory (Wikipedia)
Group Theory, Branches of Mathematics
Source Lagrange's Theorem, Group Theory (Wikipedia)

Named After

Joseph-Louis Lagrange, Mathematicians

Derived from the theorem's own name (unambiguous possessive-token match to exactly one live mathematician entity, w-bfill-g5-0924 browse backfill)

Proved By

Camille Jordan, Mathematicians

Jordan proved the theorem for all permutation groups in 1861, generalizing Lagrange's original special case.

Source Lagrange's Theorem, Group Theory (Wikipedia)
Joseph-Louis Lagrange, Mathematicians

Why this is disputed. Lagrange stated only the special case for permutations of polynomial variables; he did not prove the general theorem that bears his name.

Source Lagrange's Theorem, Group Theory (Wikipedia)
Sources
1. Lagrange's Theorem, Group Theory (Wikipedia)
Wikimedia Foundation
  • opening sentence
    That is, the order (number of elements) of every subgroup divides the order of the whole group.
  • History section
    the generalization to abstract groups came later through the work of Gauss (1801), Cauchy (1844), and finally Camille Jordan (1861), who proved it for all permutation groups
  • In Branch: Group Theory
View the Source
Dissenting Readings (1 dissenting reading)
Proved By: Joseph-Louis Lagrange

Lagrange's own 1771 paper (Reflexions sur la resolution algebrique des equations) proves only a special case, about the number of distinct polynomials produced by permuting the variables of a polynomial in n variables, always a factor of n!. The general theorem, that the order of any subgroup of a finite group divides the order of the group, was not established by Lagrange himself. It was proved for all permutation groups only in 1861, by Camille Jordan, after partial extensions by Gauss in 1801 and Cauchy in 1844. Attributing the general theorem to Lagrange by eponym overstates what Lagrange actually proved.

A dissenting reading, from a reviewerLagrange's Theorem, Group Theory (Wikipedia), Wikimedia Foundation
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