# Index of a subgroup

An infinite group *G* may have subgroups *H* of finite index (for example, the even integers inside the group of integers). Such a subgroup always contains a normal subgroup *N* (of *G*), also of finite index. In fact, if *H* has index *n*, then the index of *N* can be taken as some factor of *n*!; indeed, *N* can be taken to be the kernel of the natural homomorphism from *G* to the permutation group of the left (or right) cosets of *H*.

A special case, *n* = 2, gives the general result that a subgroup of index 2 is a normal subgroup, because the normal subgroup (*N* above) must have index 2 and therefore be identical to the original subgroup. More generally, a subgroup of index *p* where *p* is the smallest prime factor of the order of *G* (if *G* is finite) is necessarily normal, as the index of *N* divides *p*! and thus must equal *p,* having no other prime factors.

An alternative proof of the result that subgroup of index lowest prime *p* is normal, and other properties of subgroups of prime index are given in (Lam 2004).

The above considerations are true for finite groups as well. For instance, the group **O** of chiral octahedral symmetry has 24 elements. It has a dihedral D_{4} subgroup (in fact it has three such) of order 8, and thus of index 3 in **O**, which we shall call *H*. This dihedral group has a 4-member D_{2} subgroup, which we may call *A*. Multiplying on the right any element of a right coset of *H* by an element of *A* gives a member of the same coset of *H* (*Hca = Hc*). *A* is normal in **O**. There are six cosets of *A*, corresponding to the six elements of the symmetric group S_{3}. All elements from any particular coset of *A* perform the same permutation of the cosets of *H*.

On the other hand, the group T_{h} of pyritohedral symmetry also has 24 members and a subgroup of index 3 (this time it is a D_{2h} prismatic symmetry group, see point groups in three dimensions), but in this case the whole subgroup is a normal subgroup. All members of a particular coset carry out the same permutation of these cosets, but in this case they represent only the 3-element alternating group in the 6-member S_{3} symmetric group.

Normal subgroups of prime power index are kernels of surjective maps to *p*-groups and have interesting structure, as described at Focal subgroup theorem: Subgroups and elaborated at focal subgroup theorem.

There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class:

As these are weaker conditions on the groups *K,* one obtains the containments

These groups have important connections to the Sylow subgroups and the transfer homomorphism, as discussed there.

An elementary observation is that one cannot have exactly 2 subgroups of index 2, as the complement of their symmetric difference yields a third. This is a simple corollary of the above discussion (namely the projectivization of the vector space structure of the elementary abelian group

and further, *G* does not act on this geometry, nor does it reflect any of the non-abelian structure (in both cases because the quotient is abelian).

However, it is an elementary result, which can be seen concretely as follows: the set of normal subgroups of a given index *p* form a projective space, namely the projective space