பயனர்:Booradleyp/test

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. (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,

|\mathbf {Z} :n\mathbf {Z} |=n

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:

|G:H|={\frac {|G|}{|H|}}.

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

Properties தொகு

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

|G:K|=|G:H|\,|H:K|.

If H and K are subgroups of G, then

|G:H\cap K|\leq |G:H|\,|G:K|,

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

|H:H\cap K|\leq |G:K|,

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:

|G:\operatorname {ker} \;\varphi |=|\operatorname {im} \;\varphi |.

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:

|Gx|=|G:G_{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:

|G:\operatorname {Core} (H)|\leq |G:H|!

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.

Examples தொகு

The alternating group $A_{n}$ has index 2 in the symmetric group $S_{n},$ and thus is normal.
The special orthogonal group SO(n) has index 2 in the orthogonal group O(n), and thus is normal.
The free abelian group Z ⊕ Z has three subgroups of index 2, namely

\{(x,y)\mid x{\text{ is even}}\},\quad \{(x,y)\mid y{\text{ is even}}\},\quad {\text{and}}\quad \{(x,y)\mid x+y{\text{ is even}}\}

.

More generally, if p is prime then Zⁿ has (pⁿ − 1) / (p − 1) subgroups of index p, corresponding to the pⁿ − 1 nontrivial homomorphisms Zⁿ → Z/pZ.^{[சான்று தேவை]}
Similarly, the free group F_n has pⁿ − 1 subgroups of index p.
The infinite dihedral group has a cyclic subgroup of index 2, which is necessarily normal.

Infinite index தொகு

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 தொகு

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 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.

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).

மேற்கோள்கள் தொகு

Lam, T. Y. (2004), "On Subgroups of Prime Index", The American Mathematical Monthly, 111 (3): 256–258, JSTOR 4145135, alternative download {{citation}}: External link in |postscript= (help); Unknown parameter |month= ignored (|date= suggested) (help)CS1 maint: postscript (link)