Editing Lagrange's theorem (group theory) (section)

{{short description|The order of a subgroup of a finite group G divides the order of G}}
{{other uses|Lagrange's theorem (disambiguation)}}
{{Group theory sidebar |Finite}}
[[File:Left cosets of Z 2 in Z 8.svg|thumb|G is the group <math>\mathbb{Z}/8\mathbb{Z}</math>, the [[Integers modulo n|integers mod 8]] under addition. The subgroup H contains only 0 and 4, and is isomorphic to <math>\mathbb{Z}/2\mathbb{Z}</math>. There are four left cosets of H: H itself, 1+H, 2+H, and 3+H (written using additive notation since this is an [[Abelian group|additive group]]). Together they partition the entire group G into equal-size, non-overlapping sets. Thus the [[Index of a subgroup|index]] [G : H] is 4.]]
In the [[mathematics|mathematical]] field of [[group theory]], '''Lagrange's theorem''' states that if H is a subgroup of any [[finite group]] {{mvar|G}}, then <math>|H|</math> is a divisor of <math>|G|</math>, i.e. the [[order of a group|order]] (number of elements) of every [[subgroup]] H divides the order of group G. 

The theorem is named after [[Joseph-Louis Lagrange]]. The following variant states that for a subgroup <math>H</math> of a finite group <math>G</math>, not only is <math>|G|/|H|</math> an integer, but its value is the [[index of a subgroup|index]] <math>[G:H]</math>, defined as the number of left [[coset]]s of <math>H</math> in <math>G</math>.
{{math_theorem|Lagrange's theorem|If {{mvar|H}} is a subgroup of a group {{mvar|G}}, then <math>\left|G\right| = \left[G : H\right] \cdot \left|H\right|.</math>}}
This variant holds even if <math>G</math> is infinite, provided that <math>|G|</math>, <math>|H|</math>, and <math>[G:H]</math> are interpreted as [[cardinal number]]s.