Lagrange's theorem (group theory)
Teorema de Lagrange (teoría de grupos)
- Date
- 1771 (generalizado 1801-1861) · confidence medium
- Place
- Berlin, Prussia (present-day Germany)
- Credited to
- Joseph-Louis Lagrange; generalized by Gauss, Cauchy and Jordan
- Field
- Knowledge & symbols
What was discovered
In 1771, working at the Berlin Academy on the solvability of algebraic equations, Lagrange proved that the number of distinct polynomials obtained by permuting the variables of an n-variable polynomial in every possible way always divides n!; the result, before group theory existed as such, was later generalized into the modern theorem.
Lagrange did not speak of abstract groups, which did not exist as a concept in 1771: he proved the result about permutations of a polynomial's variables. Gauss proved it in 1801 for the multiplicative group modulo a prime, Cauchy in 1844 for the symmetric group, and Camille Jordan in 1861 for any permutation group; the current abstract form came with late-19th-century group theory.
What it led to
- Cauchy and Jordan's theory of permutation groups
- The modern theorem: the order of a subgroup always divides the order of the group