Editing Index of a subgroup (section)

{{Short description|Mathematics group theory concept}}
In [[mathematics]], specifically [[group theory]], the '''index''' of a [[subgroup]] ''H'' in a group ''G'' is the 
number of left [[Coset|cosets]] of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''.
The index is denoted <math>|G:H|</math> or <math>[G:H]</math> or <math>(G:H)</math>.
Because ''G'' is the disjoint union of the left cosets and because each left coset has the same [[cardinality|size]] as ''H'', the index is related to the [[order (group theory)|orders]] of the two groups by the formula
:<math>|G| = |G:H| |H|</math>
(interpret the quantities as [[cardinal numbers]] if some of them are infinite).
Thus the index <math>|G:H|</math> measures the "relative sizes" of ''G'' and ''H''.

For example, let <math>G = \Z</math> be the group of integers under [[addition]], and let <math>H = 2\Z</math> be the subgroup consisting of the [[Parity (mathematics)|even integers]].  Then <math>2\Z</math>  has two cosets in <math>\Z</math>, namely the set of even integers and the set of odd integers, so the index <math>|\Z:2\Z|</math> is 2.  More generally, <math>|\Z:n\Z| = n</math> for any positive integer ''n''.

When ''G'' is [[finite group|finite]], the formula may be written as <math>|G:H| = |G|/|H|</math>, and it implies
[[Lagrange's theorem (group theory)|Lagrange's theorem]] that <math>|H|</math> divides <math>|G|</math>.

When ''G'' is infinite, <math>|G:H|</math> is a nonzero [[cardinal number]] that may be finite or infinite.
For example, <math>|\Z:2\Z| = 2</math>, but <math>|\R:\Z|</math> is infinite.

If ''N'' is a [[normal subgroup]] of ''G'', then <math>|G:N|</math> is equal to the order of the [[quotient group]] <math>G/N</math>, since the underlying set of <math>G/N</math> is the set of cosets of ''N'' in ''G''.