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
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).
be the group of integers under addition, and let
for any positive integer n. When G is finite, the formula may be written as
is a nonzero cardinal number that may be finite or infinite.
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. A subgroup H of finite index in a group G (finite or infinite) always contains a normal subgroup N (of G), also of finite index.
and a multiple 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. Let us explain this in more detail, using right cosets: The elements of G that leave all cosets the same form a group.
This is because the coset Hc is the same as Hca, so Hcb = Hcab.
Finally, if for some c ∈ G and a ∈ A we have ca = xc, then for any d ∈ G dca = dxc, but also dca = hdc for some h ∈ H (by the definition of A), so hd = dx.
Since this is true for any d, x must be a member of A, so ca = xc implies that cac−1 ∈ A and therefore A is a normal subgroup.
The index of the normal subgroup not only has to be a divisor of n!, but must satisfy other criteria as well.
Since the normal subgroup is a subgroup of H, its index in G must be n times its index inside H. Its index in G must also correspond to a subgroup of the symmetric group Sn, the group of permutations of n objects.
So for example if n is 5, the index cannot be 15 even though this divides 5!, because there is no subgroup of order 15 in S5.
In the case of n = 2 this gives the rather obvious result that a subgroup H of index 2 is a normal subgroup, because the normal subgroup of H must have index 2 in G and therefore be identical to H. (We can arrive at this fact also by noting that all the elements of G that are not in H constitute the right coset of H and also the left coset, so the two are identical.)
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!
An alternative proof of the result that a subgroup of index lowest prime p is normal, and other properties of subgroups of prime index are given in (Lam 2004).
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).
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.
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 In detail, the space of homomorphisms from G to the (cyclic) group of order p,
(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
As a consequence, the number of normal subgroups of index p is for some k;
corresponds to no normal subgroups of index p. Further, given two distinct normal subgroups of index p, one obtains a projective line consisting of
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