Editing Subgroup (section)

==Cosets and Lagrange's theorem==
{{Main|Coset|Lagrange's theorem (group theory)}}
Given a subgroup {{mvar|H}} and some {{mvar|a}} in {{mvar|G}}, we define the '''left [[coset]]''' {{math|1=''aH'' = {''ah'' : ''h'' in ''H''}.}} Because {{mvar|a}} is invertible, the map {{math|φ : ''H'' → ''aH''}}  given by {{math|1=φ(''h'') = ''ah''}} is a [[bijection]]. Furthermore, every element of {{mvar|G}} is contained in precisely one left coset of {{mvar|H}}; the left cosets are the equivalence classes corresponding to the [[equivalence relation]] {{math|''a''<sub>1</sub> ~ ''a''<sub>2</sub>}} [[if and only if]] {{tmath|a_1^{-1}a_2}} is in {{mvar|H}}. The number of left cosets of {{mvar|H}} is called the [[index of a subgroup|index]] of {{mvar|H}} in {{mvar|G}} and is denoted by {{math|[''G'' : ''H'']}}.

[[Lagrange's theorem (group theory)|Lagrange's theorem]] states that for a finite group {{mvar|G}} and a subgroup {{mvar|H}}, 
: <math> [ G : H ] = { |G| \over |H| }</math>
where {{mvar|{{abs|G}}}} and  {{mvar|{{abs|H}}}} denote the [[order (group theory)|order]]s of {{mvar|G}} and {{mvar|H}}, respectively. In particular, the order of every subgroup of {{mvar|G}} (and the order of every element of {{mvar|G}}) must be a [[divisor]] of {{mvar|{{abs|G}}}}.<ref>See a [https://www.youtube.com/watch?v=TCcSZEL_3CQ didactic proof in this video].</ref>{{sfn|Dummit|Foote|2004|p=90}}

'''Right cosets''' are defined analogously: {{math|1=''Ha'' = {''ha'' : ''h'' in ''H''}.}} They are also the equivalence classes for a suitable equivalence relation and their number is equal to {{math|[''G'' : ''H'']}}.

If {{math|1=''aH'' = ''Ha''}} for every {{mvar|a}} in {{mvar|G}}, then {{mvar|H}} is said to be a [[normal subgroup]]. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if {{mvar|p}} is the lowest prime dividing the order of a finite group {{mvar|G}}, then any subgroup of index {{mvar|p}} (if such exists) is normal.