Index of a subgroup
Encyclopedia
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] .
Formally, the index of H in G is defined as the number of coset
s of H in G. (The number of left cosets of H in G is always equal to the number of right cosets.) For example, let Z be the group of integers under addition
, and let 2Z be the subgroup of Z consisting of the even integers. Then 2Z has two cosets in Z (namely the even integers and the odd integers), so the index of 2Z in Z is two. In general,
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 cardinal number
. It may however be finite, that is, a positive integer, as the example above shows.
If G and H are finite group
s, 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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, the index of a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
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
Formally, the index of H in G is defined as the number of coset
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
s of H in G. (The number of left cosets of H in G is always equal to the number of right cosets.) For example, let Z be the group of integers under addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
, and let 2Z be the subgroup of Z consisting of the even integers. Then 2Z has two cosets in Z (namely the even integers and the odd integers), so the index of 2Z in Z is two. In general,
for any positive integer n.
If N is a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of G, then the index of N in G is also equal to the order of the quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
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 cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
. It may however be finite, that is, a positive integer, as the example above shows.
If G and H are finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s, then the index of H in G is equal to the quotient
Quotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
of the orders
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
of the two groups:
This is Lagrange's theorem
Lagrange's theorem (group theory)
Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange....
, and in this case the quotient is necessarily a positive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
.
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 homomorphismHomomorphismIn abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
, then the index of the kernelKernel (algebra)In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
of φ in G is equal to the order of the image:
- Let G be a group actingGroup actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
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:
- Let G be a group acting
- If G and H are groups and φ: G → H is a homomorphism
- This is known as the orbit-stabilizer theorem.
- As a special case of the orbit-stabilizer theorem, the number of conjugatesConjugacy classIn mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
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 coreCore (group)In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the p-core of a group.-Definition:...
of H satisfies the following inequality:
- As a special case of the orbit-stabilizer theorem, the number of conjugates
- where ! denotes the factorialFactorialIn mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
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 groupSimple groupIn mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
of non-prime order, or more generally any perfect groupPerfect groupIn mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients...
.
Examples
- The alternating group
has index 2 in the symmetric groupSymmetric groupIn mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
and thus is normal. - The special orthogonal group SO(n) has index 2 in the orthogonal groupOrthogonal groupIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
O(n), and thus is normal. - The free abelian groupFree abelian groupIn abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
Z ⊕ Z has three subgroups of index 2, namely
-
.
- More generally, if p is primePrime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
then Zn has (pn − 1) / (p − 1) subgroups of index p, corresponding to the pn − 1 nontrivial homomorphismHomomorphismIn abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
s Zn → Z/pZ. - Similarly, the free groupFree groupIn mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
Fn has pn − 1 subgroups of index p. - The infinite dihedral groupInfinite dihedral groupIn mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups.-Definition:...
has a cyclic subgroupCyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
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 numberCardinal numberIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
. For example, the index of H in G may be countableCountable setIn mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
or uncountableUncountable setIn mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.-Characterizations:There...
, 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 subgroupNormal subgroupIn abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
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!.
A special case, n = 2, gives the general result that a subgroup of index 2 is a normal subgroup, because the normal group (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.
This result is generally proven using group actions; 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 .
Proof
This can be seen more concretely, by considering the permutation action of G on left cosets of H when multiplying them on the right by elements of G (or, equally, multiplying right cosets on the left). This provides a quotient group of G, the image of this permutation representation, which is a subgroup of the symmetric groupSymmetric groupIn mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
on n elements.
Let us explain this now in more detail.
The elements of G that leave all cosets the same form a group.
(If Hca ⊂ Hc ∀ c ∈ G and likewise Hcb ⊂ Hc ∀ c ∈ G, then Hcab ⊂ Hc ∀ c ∈ G. If h1ca = h2c for all c ∈ G (with h1, h2 ∈ H) then h2ca−1 = h1c, so Hca−1 ⊂ Hc.)
Let us call this group A. Let B be the set of elements of G which perform a given permutation on the cosets of H. Then the cardinality (size) of B is equal to the cardinality of A, and in fact B is a right coset of A.
(If cb1 = d and cb2 = hd (a member of the same coset as d), then cb1b2−1 = db2−1 = h−1c ∈ Hc. Since this is the case for any b2 and for any c (with appropriate d), b1b2−1 ∈ A and the size of B is less than or equal to the size of A. Conversely, Hcb1 = Hcab1, and since the left-hand side is in Hd then so is the right-hand side: Hcab1 ⊂ Hcd, showing that for any element of A there is a different element of B, and thus the size of A is less than or equal to the size of B.)
Since the number of possible permutations of cosets is finite, namely n! (assuming H is of finite index n), then there can only be a finite number of sets like B. If G is infinite, then all such sets are therefore infinite. The set of these sets forms a group isomorphic to a subset of the group of permutations, so the number of these sets must divide n!. Finally, if for some c ∈ G and a ∈ A we have ca = xc, then for any d ∈ G dca = hdc for some h ∈ H, but also dca = dxc, so hd = dx. Since this is true for any d, x must be a member of A, so ca = xc implies that A is a normal subgroupNormal subgroupIn abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
.
Examples
The above considerations are true for finite groups as well. For instance, the group O of chiral octahedral symmetryOctahedral symmetry150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...
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 groupSymmetric groupIn mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
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 dimensionsPoint groups in three dimensionsIn geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, the group...
), 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 powerPrime powerIn mathematics, a prime power is a positive integer power of a prime number.For example: 5=51, 9=32 and 16=24 are prime powers, while6=2×3, 15=3×5 and 36=62=22×32 are not...
index are kernels of surjective maps to p-groupsP-groupIn mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...
and have interesting structure, as described at Focal subgroup theorem: Subgroups and elaborated at focal subgroup theoremFocal subgroup theoremIn abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in and is the "first major application of the transfer" according to . The focal subgroup theorem relates the ideas of transfer and...
.
There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class:- Ep(G) is the intersection of all index p normal subgroups; G/Ep(G) is an elementary abelian groupElementary Abelian groupIn group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order p, where p is a prime; in particular it is a p-group....
, and is the largest elementary abelian p-group onto which G surjects. - Ap(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/Ap(G) is the largest abelian p-group (not necessarily elementary) onto which G surjects. - Op(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/Op(G) is the largest p-group (not necessarily abelian) onto which G surjects. Op(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 complementComplement (set theory)In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
of their symmetric differenceSymmetric differenceIn mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....
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 spaceProjective spaceIn mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
, 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 fieldFinite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
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 lineProjective lineIn mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
consisting of
such subgroups.
For
the symmetric differenceSymmetric differenceIn mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....
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.
External links
- More generally, if p is prime