Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup "H" in a group "G" is the 
number of left cosets of "H" in "G", or equivalently, the number of right cosets of "H" in "G".
The index is denoted formula_1 or formula_2 or formula_3.
Because "G" is the disjoint union of the left cosets and because each left coset has the same size as "H", the index is related to the orders of the two groups by the formula
(interpret the quantities as cardinal numbers if some of them are infinite).
Thus the index formula_1 measures the "relative sizes" of "G" and "H".
For example, let formula_6 be the group of integers under addition, and let formula_7 be the subgroup consisting of the even integers. Then formula_8 has two cosets in formula_9, namely the set of even integers and the set of odd integers, so the index formula_10 is 2. More generally, formula_11 for any positive integer "n".
When "G" is finite, the formula may be written as formula_12, and it implies
Lagrange's theorem that formula_13 divides formula_14.
When "G" is infinite, formula_1 is a nonzero cardinal number that may be finite or infinite.
For example, formula_16, but formula_17 is infinite.
If "N" is a normal subgroup of "G", then formula_18 is equal to the order of the quotient group formula_19, since the underlying set of formula_19 is the set of cosets of "N" in "G".
Infinite index.
If "H" has an infinite number of cosets in "G", then the index of "H" in "G" is said to be infinite. In this case, the index formula_1 is actually a cardinal number. For example, the index of "H" in "G" may be countable or uncountable, depending on whether "H" has a countable number of cosets in "G". Note that the index of "H" is at most the order of "G," which is realized for the trivial subgroup, or in fact any subgroup "H" of infinite cardinality less than that of "G."
Finite index.
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 .
Examples.
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 D4 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 D2 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 S3. All elements from any particular coset of "A" perform the same permutation of the cosets of "H".
On the other hand, the group Th of pyritohedral symmetry also has 24 members and a subgroup of index 3 (this time it is a D2h 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 S3 symmetric group.
Normal subgroups of prime power index.
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.
Geometric structure.
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 
In detail, the space of homomorphisms from "G" to the (cyclic) group of order "p," formula_57 is a vector space over the finite field formula_58 A non-trivial such map has as kernel a normal subgroup of index "p," and multiplying the map by an element of formula_59 (a non-zero number mod "p") does not change the kernel; thus one obtains a map from 
to normal index "p" subgroups. Conversely, a normal subgroup of index "p" determines a non-trivial map to formula_61 up to a choice of "which coset maps to formula_62 which shows that this map is a bijection.
As a consequence, the number of normal subgroups of index "p" is 
for some "k;" formula_64 corresponds to no normal subgroups of index "p". Further, given two distinct normal subgroups of index "p," one obtains a projective line consisting of formula_65 such subgroups.
For formula_66 the symmetric difference of two distinct index 2 subgroups (which are necessarily normal) gives the third point on the projective line containing these subgroups, and a group must contain formula_67 index 2 subgroups – it cannot contain exactly 2 or 4 index 2 subgroups, for instance.