Lagrange’s theorem on finite groups

mathematics

Also known as: Smith-Helmholtz theorem

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June 30, 2026

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Lagrange's theorem, in the context of finite groups, is a fundamental concept in abstract algebra. It states that if G is a finite group and H is a subgroup of G, then the order of H (the number of elements in H) must divide the order of G. This means that the size of any subgroup is always a factor of the size of the larger group it belongs to.1

This theorem was a significant contribution by mathematician Joseph-Louis Lagrange, who also made important advancements in number theory and mechanics. The study of groups, known as group theory, has broad applications in various fields, including physics, chemistry, and computer science.21

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optics
Magnification: the optical invariant
This theorem has been named after the French scientist Joseph-Louis Lagrange, although it is sometimes called the Smith-Helmholtz theorem, after Robert Smith, an English scientist, and Hermann Helmholtz, a German scientist; the product (hnu) is often known as the optical invariant. As it is easy to…
Read More
This theorem has been named after the French scientist Joseph-Louis Lagrange, although it is sometimes called the Smith-Helmholtz theorem, after Robert Smith, an English scientist, and Hermann Helmholtz, a German scientist; the product (hnu) is often known as the optical invariant. As it is easy to…
Read More