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

== Extension ==
Lagrange's theorem can be extended to the equation of indexes between three subgroups of {{mvar|G}}.<ref>{{mathworld |author=Bray, Nicolas |title=Lagrange's Group Theorem |id=LagrangesGroupTheorem|mode=cs2}}</ref>
{{math_theorem|Extension of Lagrange's theorem|If {{mvar|H}} is a subgroup of {{mvar|G}} and {{mvar|K}} is a subgroup of {{mvar|H}}, then
:<math>[G:K] = [G:H]\,[H:K].</math>}}
{{math_proof|Let {{mvar|S}} be a set of coset representatives for {{mvar|K}} in {{mvar|H}}, 
so <math>H = \bigsqcup_{s \in S} sK</math> (disjoint union), and <math>|S|=[H:K]</math>.
For any <math>a \in G</math>, left-multiplication-by-{{mvar|a}} is a bijection <math>G \to G</math>, 
so <math>aH = \bigsqcup_{s \in S} asK</math>.  
Thus each left coset of {{mvar|H}} decomposes into <math>[H:K]</math> left cosets of {{mvar|K}}.
Since {{mvar|G}} decomposes into <math>[G:H]</math> left cosets of {{mvar|H}},
each of which decomposes into <math>[H:K]</math> left cosets of {{mvar|K}},
the total number <math>[G:K]</math> of left cosets of {{mvar|K}} in {{mvar|G}} is <math>[G:H][H:K]</math>.
}}
If we take {{math|''K'' {{=}} {{mset|''e''}}}} ({{mvar|e}} is the identity element of {{mvar|G}}), then {{math|[''G'' : {{mset|''e''}}] {{=}} {{!}}''G''{{!}}}} and {{math|[''H'' : {{mset|''e''}}] {{=}} {{!}}''H''{{!}}}}.  Therefore, we can recover the original equation {{math|{{!}}''G''{{!}} {{=}} [''G'' : ''H''] {{!}}''H''{{!}}}}.