Editing Index of a subgroup (section)

==Examples==
* The [[alternating group]] <math>A_n</math> has index 2 in the [[symmetric group]] <math>S_n,</math> and thus is normal.
* The [[special orthogonal group]] <math>\operatorname{SO}(n)</math> has index 2 in the [[orthogonal group]] <math>\operatorname{O}(n)</math>, and thus is normal.
* The [[free abelian group]] <math>\Z\oplus \Z</math> has three subgroups of index 2, namely
::<math>\{(x,y) \mid x\text{ is even}\},\quad \{(x,y) \mid y\text{ is even}\},\quad\text{and}\quad
\{(x,y) \mid x+y\text{ is even}\}</math>.
* More generally, if ''p'' is [[prime number|prime]] then <math>\Z^n</math> has <math>(p^n-1)/(p-1)</math> subgroups of index ''p'', corresponding to the <math>(p^n-1)</math> nontrivial [[homomorphism]]s <math>\Z^n \to \Z/p\Z</math>.{{Citation needed|date=January 2010}}
* Similarly, the [[free group]] <math>F_n</math> has <math>(p^n-1)/(p - 1)</math>  subgroups of index ''p''.
* The [[infinite dihedral group]] has a [[cyclic group|cyclic subgroup]] of index 2, which is necessarily normal.