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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'').
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| 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. To generalize,
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| :<math>|\mathbf{Z}:n\mathbf{Z}| = n</math>
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| for any positive integer ''n''. | |
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| 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''.
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| 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.
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| 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:
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| :<math>|G:H| = \frac{|G|}{|H|}.</math>
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| This is [[Lagrange's theorem (group theory)|Lagrange's theorem]], and in this case the quotient is necessarily a positive [[integer]].
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| ==Properties==
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| * If ''H'' is a subgroup of ''G'' and ''K'' is a subgroup of ''H'', then
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| ::<math>|G:K| = |G:H|\,|H:K|.</math>
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| * If ''H'' and ''K'' are subgroups of ''G'', then
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| ::<math>|G:H\cap K| \le |G : H|\,|G : K|,</math>
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| :with equality if ''HK'' = ''G''. (If |''G'' : ''H'' ∩ ''K''| is finite, then equality holds if and only if ''HK'' = ''G''.)
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| * Equivalently, if ''H'' and ''K'' are subgroups of ''G'', then
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| ::<math>|H:H\cap K| \le |G:K|,</math>
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| :with equality if ''HK'' = ''G''. (If |''H'' : ''H'' ∩ ''K''| is finite, then equality holds if and only if ''HK'' = ''G''.)
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| * 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:
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| ::<math>|G:\operatorname{ker}\;\varphi|=|\operatorname{im}\;\varphi|.</math>
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| * 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'':
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| ::<math>|Gx| = |G:G_x|.\!</math>
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| :This is known as the [[orbit-stabilizer theorem]].
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| * 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''.
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| * 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''.
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| * If ''H'' is a subgroup of ''G'', the index of the [[core (group)|normal core]] of ''H'' satisfies the following inequality:
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| ::<math>|G:\operatorname{Core}(H)| \le |G:H|!</math>
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| :where ! denotes the [[factorial]] function; this is discussed further [[#Finite index|below]].
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| :* 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.
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| :* 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]].
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| ==Examples==
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| * The [[alternating group]] <math>A_n</math> has index 2 in the [[symmetric group]] <math>S_n,</math> and thus is normal.
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| * The [[special orthogonal group]] ''SO''(''n'') has index 2 in the [[orthogonal group]] ''O''(''n''), and thus is normal.
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| * The [[free abelian group]] '''Z''' ⊕ '''Z''' has three subgroups of index 2, namely
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| ::<math>\{(x,y) \mid x\text{ is even}\},\quad \{(x,y) \mid y\text{ is even}\},\quad\text{and}\quad
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| \{(x,y) \mid x+y\text{ is even}\}</math>.
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| * 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}}
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| * Similarly, the [[free group]] ''F''<sub>''n''</sub> has ''p''<sup>''n''</sup> − 1 subgroups of index ''p''.
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| * The [[infinite dihedral group]] has a [[cyclic group|cyclic subgroup]] of index 2, which is necessarily normal.
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| ==Infinite index==
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| 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.''
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| ==Finite index==
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| 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''.
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| 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.
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| 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}}.
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| === Examples ===
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| 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 symmetry|dihedral]] D<sub>4</sub> 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<sub>2</sub> 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<sub>3</sub>. All elements from any particular coset of ''A'' perform the same permutation of the cosets of ''H''.
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| On the other hand, the group T<sub>h</sub> of [[pyritohedral symmetry]] also has 24 members and a subgroup of index 3 (this time it is a D<sub>2h</sub> [[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<sub>3</sub> symmetric group.
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| ==Normal subgroups of prime power index==
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| Normal subgroups of [[prime power]] index are kernels of surjective maps to [[p-group|''p''-groups]] and have interesting structure, as described at [[Focal subgroup theorem#Subgroups|Focal subgroup theorem: Subgroups]] and elaborated at [[focal subgroup theorem]].
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| There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class:
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| * '''E'''<sup>''p''</sup>(''G'') is the intersection of all index ''p'' normal subgroups; ''G''/'''E'''<sup>''p''</sup>(''G'') is an [[elementary abelian group]], and is the largest elementary abelian ''p''-group onto which ''G'' surjects.
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| * '''A'''<sup>''p''</sup>(''G'') is the intersection of all normal subgroups ''K'' such that ''G''/''K'' is an abelian ''p''-group (i.e., ''K'' is an index <math>p^k</math> normal subgroup that contains the derived group <math>[G,G]</math>): ''G''/'''A'''<sup>''p''</sup>(''G'') is the largest abelian ''p''-group (not necessarily elementary) onto which ''G'' surjects.
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| * '''O'''<sup>''p''</sup>(''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 <math>p^k</math> normal subgroup): ''G''/'''O'''<sup>''p''</sup>(''G'') is the largest ''p''-group (not necessarily abelian) onto which ''G'' surjects. '''O'''<sup>''p''</sup>(''G'') is also known as the {{anchor|p-residual subgroup}}'''''p''-residual subgroup'''.
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| As these are weaker conditions on the groups ''K,'' one obtains the containments
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| :<math>\mathbf{E}^p(G) \supseteq \mathbf{A}^p(G) \supseteq \mathbf{O}^p(G).</math>
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| These groups have important connections to the [[Sylow subgroup]]s and the transfer homomorphism, as discussed there.
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| === Geometric structure ===
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| An elementary observation is that one cannot have exactly 2 subgroups of index 2, as the [[Complement (set theory)|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
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| :<math>G/\mathbf{E}^p(G) \cong (\mathbf{Z}/p)^k</math>),
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| 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).
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| 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
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| :<math>\mathbf{P}(\operatorname{Hom}(G,\mathbf{Z}/p)).</math>
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| In detail, the space of homomorphisms from ''G'' to the (cyclic) group of order ''p,'' <math>\operatorname{Hom}(G,\mathbf{Z}/p),</math> is a vector space over the [[finite field]] <math>\mathbf{F}_p = \mathbf{Z}/p.</math> A non-trivial such map has as kernel a normal subgroup of index ''p,'' and multiplying the map by an element of <math>(\mathbf{Z}/p)^\times</math> (a non-zero number mod ''p'') does not change the kernel; thus one obtains a map from
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| :<math>\mathbf{P}(\operatorname{Hom}(G,\mathbf{Z}/p)) := (\operatorname{Hom}(G,\mathbf{Z}/p))\setminus\{0\})/(\mathbf{Z}/p)^\times</math>
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| to normal index ''p'' subgroups. Conversely, a normal subgroup of index ''p'' determines a non-trivial map to <math>\mathbf{Z}/p</math> up to a choice of "which coset maps to <math>1 \in \mathbf{Z}/p,</math> which shows that this map is a bijection.
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| As a consequence, the number of normal subgroups of index ''p'' is
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| :<math>(p^{k+1}-1)/(p-1)=1+p+\cdots+p^k</math>
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| for some ''k;'' <math>k=-1</math> corresponds to no normal subgroups of index ''p''. Further, given two distinct normal subgroups of index ''p,'' one obtains a [[projective line]] consisting of <math>p+1</math> such subgroups.
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| For <math>p=2,</math> 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 <math>0,1,3,7,15,\ldots</math> index 2 subgroups – it cannot contain exactly 2 or 4 index 2 subgroups, for instance.
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| ==See also==
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| * [[Virtually]]
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| * [[Codimension]]
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| ==References==
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| {{Refimprove|date=January 2010}}
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| {{reflist}}
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| {{Refbegin}}
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| * {{ Citation | title = On Subgroups of Prime Index | first = T. Y. | last = Lam | journal = [[The American Mathematical Monthly]] | volume = 111 | number = 3 |date=March 2004 | pages = 256–258 | jstor = 4145135 | postscript =, [http://math.berkeley.edu/~lam/html/index-p.ps alternative download] }}
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| {{Refend}}
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| == External links ==
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| * {{PlanetMath | urlname = NormalityOfSubgroupsOfPrimeIndex | title = Normality of subgroups of prime index }}
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| * "[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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| {{DEFAULTSORT:Index Of A Subgroup}}
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| [[Category:Group theory]]
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