# Index of a subgroup

In mathematics, specifically group theory, the **index** of a subgroup *H* in a group *G* is the "relative size" of *H* in *G*: equivalently, the number of "copies" (cosets) of *H* that fill up *G*. For example, if *H* has index 2 in *G*, then intuitively half of the elements of *G* lie in *H*. The index of *H* in *G* is usually denoted |*G* : *H*| or [*G* : *H*] or (*G*:*H*).

Formally, the index of *H* in *G* is defined as the number of cosets of *H* in *G*. (It is always the case that the number of left cosets of *H* in *G* is equal to the number of right cosets.) For example, let **Z** be the group of integers under addition, and let 2**Z** be the subgroup of **Z** consisting of the even integers. Then 2**Z** has two cosets in **Z** (namely the even integers and the odd integers), so the index of 2**Z** in **Z** is two. To generalize,

for any positive integer *n*.

If *N* is a normal subgroup of *G*, then the index of *N* in *G* is also equal to the order of the quotient group *G* / *N*, since this is defined in terms of a group structure on the set of cosets of *N* in *G*.

If *G* is infinite, the index of a subgroup *H* will in general be a non-zero cardinal number. It may be finite - that is, a positive integer - as the example above shows.

If *G* and *H* are finite groups, then the index of *H* in *G* is equal to the quotient of the orders of the two groups:

This is Lagrange's theorem, and in this case the quotient is necessarily a positive integer.

## Properties

- If
*H*is a subgroup of*G*and*K*is a subgroup of*H*, then

- If
*H*and*K*are subgroups of*G*, then

- with equality if
*HK*=*G*. (If |*G*:*H*∩*K*| is finite, then equality holds if and only if*HK*=*G*.)

- Equivalently, if
*H*and*K*are subgroups of*G*, then

- with equality if
*HK*=*G*. (If |*H*:*H*∩*K*| is finite, then equality holds if and only if*HK*=*G*.)

- If
*G*and*H*are groups and*φ*:*G*→*H*is a homomorphism, then the index of the kernel of*φ*in*G*is equal to the order of the image:

- Let
*G*be a group acting on a set*X*, and let*x*∈*X*. Then the cardinality of the orbit of*x*under*G*is equal to the index of the stabilizer of*x*:

- This is known as the orbit-stabilizer theorem.

- As a special case of the orbit-stabilizer theorem, the number of conjugates
*gxg*^{−1}of an element*x*∈*G*is equal to the index of the centralizer of*x*in*G*. - Similarly, the number of conjugates
*gHg*^{−1}of a subgroup*H*in*G*is equal to the index of the normalizer of*H*in*G*. - If
*H*is a subgroup of*G*, the index of the normal core of*H*satisfies the following inequality:

- where ! denotes the factorial function; this is discussed further below.
- As a corollary, if the index of
*H*in*G*is 2, or for a finite group the lowest prime*p*that divides the order of*G,*then*H*is normal, as the index of its core must also be*p,*and thus*H*equals its core, i.e., is normal. - Note that a subgroup of lowest prime index may not exist, such as in any simple group of non-prime order, or more generally any perfect group.

- As a corollary, if the index of

## 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*(*2*), and thus is normal. - The free abelian group
**Z**⊕**Z**has three subgroups of index 2, namely

- .

- More generally, if
*p*is prime then**Z**^{n}has (*p*^{n}− 1) / (*p*− 1) subgroups of index*p*, corresponding to the*p*^{n}− 1 nontrivial homomorphisms**Z**^{n}→**Z**/*p***Z**. - Similarly, the free group
*F*_{n}has*p*^{n}− 1 subgroups of index*p*. - The infinite dihedral group has a cyclic subgroup of index 2, which is necessarily normal.

## 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 |*G* : *H*| 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 (Lam 2004).

### 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 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

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:

**E**^{p}(*G*) is the intersection of all index*p*normal subgroups;*G*/**E**^{p}(*G*) is an elementary abelian group, and is the largest elementary abelian*p*-group onto which*G*surjects.**A**^{p}(*G*) is the intersection of all normal subgroups*K*such that*G*/*K*is an abelian*p*-group (i.e.,*K*is an index normal subgroup that contains the derived group ):*G*/**A**^{p}(*G*) is the largest abelian*p*-group (not necessarily elementary) onto which*G*surjects.**O**^{p}(*G*) is the intersection of all normal subgroups*K*of*G*such that*G*/*K*is a (possibly non-abelian)*p*-group (i.e.,*K*is an index normal subgroup):*G*/**O**^{p}(*G*) is the largest*p*-group (not necessarily abelian) onto which*G*surjects.**O**^{p}(*G*) is also known as the.*p*-residual subgroup

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,* is a vector space over the finite field A non-trivial such map has as kernel a normal subgroup of index *p,* and multiplying the map by an element of (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 up to a choice of "which coset maps to which shows that this map is a bijection.

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 such subgroups.

For 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 index 2 subgroups – it cannot contain exactly 2 or 4 index 2 subgroups, for instance.

## See also

## References

- Lam, T. Y. (March 2004), "On Subgroups of Prime Index",
*The American Mathematical Monthly*,**111**(3): 256–258, JSTOR 4145135, alternative download