Theorems
Lagrange's Theorem (Group Theory)
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Algebra
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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.
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Disputed
Proof YearLagrange 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. StatementFor 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 Cross-Tradition Connections
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Jordan proved the theorem for all permutation groups in 1861, generalizing Lagrange's original special case.
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.
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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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