nLab Lagrange's theorem

Idea

In group theory, Lagrange’s theorem asserts that for $G$ a finite group and $H \,\subset\, G$ a subgroup, the order $\left\vert H\right\vert$ of $H$ divides the order $\left\vert G \right\vert$ of $G$, in that the quotient

$$[G : H] \;\coloneqq\; \frac { \left\vert G \right\vert } { \left\vert H \right\vert } \;\;\; \in \; \mathbb{N}$$

is a natural number, called the index of $H$ in $G$.

Related statements

A partial converse to Lagrange’s theorem is

• Sylow's theorem

References

Named after:

• Joseph-Louis Lagrange, Réflexions sur la résolution algébrique des équations, Nouveaux mémoires de l’Académie royale des sciences et belles-lettres de Berlin, années 1770 et 1771, Œuvres complètes, tome 3, 205-421 (mathdoc)

More on the history:

• Richard L. Roth, A History of Lagrange’s Theorem on Groups, Mathematics Magazine 74 2 (2001) 99-108 (doi:10.2307/2690624, jstor:2690624)

See also:

• Wikipedia, Lagrange's theorem (group theory)