Index of a subgroup

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In mathematics, specifically group theory, the index of a subgroup H in a group G is the “relative size” of H in 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].

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

By Lagrange's theorem, this number is always a positive integer.

If G and H are infinite, then the index of H is G is defined as the number of cosets 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 equal to the order of the quotient group G / N.

Contents 1 Properties 2 Examples 3 Infinite index 4 Finite index 5 See also

[edit] 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.

[edit] Examples

• The special orthogonal group SO(n) has index 2 in the orthogonal group O(n).
• The free abelian group Z ⊕ Z has three subgroups of index 2, namely

.

• More generally, if p is prime then Zⁿ has pⁿ − 1 subgroups of index p, corresponding to the pⁿ − 1 nontrivial homomorphisms Zⁿ → Z/pZ.
• Similarly, the free group F_n has pⁿ − 1 subgroups of index p.
• The infinite dihedral group has a cyclic subgroup of index 2.

[edit] 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 |H : G| 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.

[edit] 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!. 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 kernel of this permutation representation, which is a subgroup of the symmetric group 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 = db^-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 subgroup.

A special case, n = 2, gives the general result that a subgroup of index 2 is a normal subgroup, because the normal group (A above) must have index 2 and therefore be identical to the original subgroup.

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 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 S₃ symmetric group.

[edit] See also

• Virtually
• Codimension