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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'': 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 <nowiki>[</nowiki>''G'' : ''H''<nowiki>]</nowiki> 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 2'''Z''' be the subgroup of '''Z''' consisting of the [[Parity (mathematics)|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. In general,
:<math>|\mathbf{Z}:n\mathbf{Z}| = n</math>
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 group]]s, then the index of ''H'' in ''G'' is equal to the [[quotient]] of the [[order (group theory)|orders]] of the two groups:
:<math>|G:H| = \frac{|G|}{|H|}.</math>
This is [[Lagrange's theorem (group theory)|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
::<math>|G:K| = |G:H|\,|H:K|.</math>
* If ''H'' and ''K'' are subgroups of ''G'', then
::<math>|G:H\cap K| \le |G : H|\,|G : K|,</math>
: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
::<math>|H:H\cap K| \le |G:K|,</math>
: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 (algebra)|kernel]] of ''φ'' in ''G'' is equal to the order of the image:
::<math>|G:\operatorname{ker}\;\varphi|=|\operatorname{im}\;\varphi|.</math>
* Let ''G'' be a group [[group action|acting]] on a [[set (mathematics)|set]] ''X'', and let ''x'' ∈ ''X''. Then the [[cardinality]] of the [[orbit (group theory)|orbit]] of ''x'' under ''G'' is equal to the index of the [[stabilizer subgroup|stabilizer]] of ''x'':
::<math>|Gx| = |G:G_x|.\!</math>
:This is known as the [[orbit-stabilizer theorem]].
* As a special case of the orbit-stabilizer theorem, the number of [[conjugacy class|conjugates]] ''gxg''<sup>−1</sup> of an element ''x'' ∈ ''G'' is equal to the index of the [[centralizer]] of ''x'' in ''G''.
* Similarly, the number of conjugates ''gHg''<sup>−1</sup> 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 [[core (group)|normal core]] of ''H'' satisfies the following inequality:
::<math>|G:\operatorname{Core}(H)| \le |G:H|!</math>
:where ! denotes the [[factorial]] function; this is discussed further [[#Finite index|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==
* The [[alternating group]] <math>A_n</math> has index 2 in the [[symmetric group]] <math>S_n,</math> 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
::<math>\{(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}\}</math>.
* More generally, if ''p'' is [[prime number|prime]] then '''Z'''<sup>''n''</sup> has (''p''<sup>''n''</sup> − 1) / (''p'' − 1) subgroups of index ''p'', corresponding to the ''p''<sup>''n''</sup> − 1 nontrivial [[homomorphism]]s '''Z'''<sup>''n''</sup> → '''Z'''/''p'''''Z'''.{{Citation needed|date=January 2010}}
* Similarly, the [[free group]] ''F''<sub>''n''</sub> has ''p''<sup>''n''</sup> − 1 subgroups of index ''p''.
* The [[infinite dihedral group]] has a [[cyclic group|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 set|countable]] or [[Uncountable set|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 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 {{Harv|Lam|2004}}.
==மேற்கோள்கள்==
{{reflist}}
* {{ Citation | title = On Subgroups of Prime Index | first = T. Y. | last = Lam | journal = [[The American Mathematical Monthly]] | volume = 111 | number = 3 | month = March | year = 2004 | pages = 256–258 | jstor = 4145135 | postscript =, [http://math.berkeley.edu/~lam/html/index-p.ps alternative download] }}
== வெளி இணைப்புகள் ==
* {{PlanetMath | urlname = NormalityOfSubgroupsOfPrimeIndex | title = Normality of subgroups of prime index }}
* "[http://groupprops.subwiki.org/wiki/Subgroup_of_least_prime_index_is_normal Subgroup of least prime index is normal]" at [http://groupprops.subwiki.org/wiki/Main_Page Groupprops, The Group Properties Wiki]
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