Index of A Subgroup - Examples

Examples

  • The alternating group has index 2 in the symmetric group and thus is normal.
  • The special orthogonal group SO(n) has index 2 in the orthogonal group O(n), and thus is normal.
  • The free abelian group Z ⊕ Z has three subgroups of index 2, namely
\{(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}\}.
  • More generally, if p is prime then Zn has (pn − 1) / (p − 1) subgroups of index p, corresponding to the pn − 1 nontrivial homomorphisms Zn → Z/pZ.
  • Similarly, the free group Fn has pn − 1 subgroups of index p.
  • The infinite dihedral group has a cyclic subgroup of index 2, which is necessarily normal.

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