In

_{4} 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_{2} 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_{3}. 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 S_{3} symmetric group.

^{''p''}(''G'') is the intersection of all index ''p'' normal subgroups; ''G''/E^{''p''}(''G'') is an elementary abelian group, and is the largest elementary abelian ''p''-group onto which ''G'' surjects.
* A^{''p''}(''G'') is the intersection of all normal subgroups ''K'' such that ''G''/''K'' is an abelian ''p''-group (i.e., ''K'' is an index $p^k$ normal subgroup that contains the derived group $[G,G]$): ''G''/A^{''p''}(''G'') is the largest abelian ''p''-group (not necessarily elementary) onto which ''G'' surjects.
* O^{''p''}(''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 $p^k$ normal subgroup): ''G''/O^{''p''}(''G'') is the largest ''p''-group (not necessarily abelian) onto which ''G'' surjects. O^{''p''}(''G'') is also known as the ''p''-residual subgroup.
As these are weaker conditions on the groups ''K,'' one obtains the containments
:$\backslash mathbf^p(G)\; \backslash supseteq\; \backslash mathbf^p(G)\; \backslash supseteq\; \backslash mathbf^p(G).$
These groups have important connections to the Sylow subgroups and the transfer homomorphism, as discussed there.

Subgroup of least prime index is normal

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mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, specifically group theory
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, the index of a subgroup
In group theory
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''H'' in a group ''G'' is the
number of left cosets
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''.
The index is denoted $,\; G:H,$ or $;\; href="/html/ALL/s/:H.html"\; ;"title=":H">:H$Properties

* If ''H'' is a subgroup of ''G'' and ''K'' is a subgroup of ''H'', then ::$,\; G:K,\; =\; ,\; G:H,\; \backslash ,,\; H:K,\; .$ * If ''H'' and ''K'' are subgroups of ''G'', then ::$,\; G:H\backslash cap\; K,\; \backslash le\; ,\; G\; :\; H,\; \backslash ,,\; G\; :\; K,\; ,$ :with equality if $HK=G$. (If $,\; G:H\backslash cap\; K,$ is finite, then equality holds if and only if $HK=G$.) * Equivalently, if ''H'' and ''K'' are subgroups of ''G'', then ::$,\; H:H\backslash cap\; K,\; \backslash le\; ,\; G:K,\; ,$ :with equality if $HK=G$. (If $,\; H:H\backslash cap\; K,$ is finite, then equality holds if and only if $HK=G$.) * If ''G'' and ''H'' are groups and $\backslash varphi\; \backslash colon\; G\backslash to\; H$ is a homomorphism, then the index of the kernel (algebra), kernel of $\backslash varphi$ in ''G'' is equal to the order of the image: ::$,\; G:\backslash operatorname\backslash ;\backslash varphi,\; =,\; \backslash operatorname\backslash ;\backslash varphi,\; .$ * Let ''G'' be a group Group action (mathematics), 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'': ::$,\; Gx,\; =\; ,\; G:G\_x,\; .\backslash !$ :This is known as the orbit-stabilizer theorem. * As a special case of the orbit-stabilizer theorem, the number of conjugacy class, conjugates $gxg^$ of an element $x\; \backslash in\; G$ is equal to the index of the centralizer of ''x'' in ''G''. * Similarly, the number of conjugates $gHg^$ 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: ::$,\; G:\backslash operatorname(H),\; \backslash le\; ,\; G:H,\; !$ :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., it 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 $A\_n$ has index 2 in the symmetric group $S\_n,$ and thus is normal. * The special orthogonal group $\backslash operatorname(n)$ has index 2 in the orthogonal group $\backslash operatorname(n)$, and thus is normal. * The free abelian group $\backslash Z\backslash oplus\; \backslash Z$ has three subgroups of index 2, namely ::$\backslash ,\backslash quad\; \backslash ,\backslash quad\backslash text\backslash quad\; \backslash $. * More generally, if ''p'' is prime number, prime then $\backslash Z^n$ has $(p^n-1)/(p-1)$ subgroups of index ''p'', corresponding to the $(p^n-1)$ nontrivial homomorphisms $\backslash Z^n\; \backslash to\; \backslash Z/p\backslash Z$. * Similarly, the free group $F\_n$ has $(p^n-1)$ 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 subgroup (''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 .Examples

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 DNormal subgroups of prime power index

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. There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class: * EGeometric structure

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 :$G/\backslash mathbf^p(G)\; \backslash cong\; (\backslash mathbf/p)^k$, 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 space, namely the projective space :$\backslash mathbf(\backslash operatorname(G,\backslash mathbf/p)).$ In detail, the space of homomorphisms from ''G'' to the (cyclic) group of order ''p,'' $\backslash operatorname(G,\backslash mathbf/p),$ is a vector space over the finite field $\backslash mathbf\_p\; =\; \backslash mathbf/p.$ A non-trivial such map has as kernel a normal subgroup of index ''p,'' and multiplying the map by an element of $(\backslash mathbf/p)^\backslash times$ (a non-zero number mod ''p'') does not change the kernel; thus one obtains a map from :$\backslash mathbf(\backslash operatorname(G,\backslash mathbf/p))\; :=\; (\backslash operatorname(G,\backslash mathbf/p))\backslash setminus\backslash )/(\backslash mathbf/p)^\backslash times$ to normal index ''p'' subgroups. Conversely, a normal subgroup of index ''p'' determines a non-trivial map to $\backslash mathbf/p$ up to a choice of "which coset maps to $1\; \backslash in\; \backslash mathbf/p,$ which shows that this map is a bijection. As a consequence, the number of normal subgroups of index ''p'' is :$(p^-1)/(p-1)=1+p+\backslash cdots+p^k$ for some ''k;'' $k=-1$ corresponds to no normal subgroups of index ''p''. Further, given two distinct normal subgroups of index ''p,'' one obtains a projective line consisting of $p+1$ such subgroups. For $p=2,$ 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 $0,1,3,7,15,\backslash ldots$ index 2 subgroups – it cannot contain exactly 2 or 4 index 2 subgroups, for instance.See also

* Virtually * CodimensionReferences

*External links

* *Subgroup of least prime index is normal

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