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 _{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 _{2h} _{3} symmetric group.

^{''p''}(''G'') is the intersection of all index ''p'' normal subgroups; ''G''/E^{''p''}(''G'') is an ^{''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 $;\; href="/html/ALL/s/,G.html"\; ;"title=",G">,G$

Subgroup of least prime index is normal

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mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, specifically group theory
In 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 ...

, the index of a subgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation âˆ—, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation âˆ—. More precisely ...

''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 $;\; href="/html/ALL/s/:H.html"\; ;"title=":H">:H$size
Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to linear dimensions (length, width, height, diameter, perimeter), area, or volume. ...

as ''H'', the index is related to the orders
Orders is a surname
In some cultures, a surname, family name, or last name is the portion of one's personal name that indicates their family, tribe or community.
Practices vary by culture. The family name may be placed at either the start of ...

of the two groups by the formula
:$,\; G,\; =\; ,\; G:H,\; ,\; H,$
(interpret the quantities as cardinal numbers
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

if some of them are infinite).
Thus the index $,\; G:H,$ measures the "relative sizes" of ''G'' and ''H''.
For example, let $G\; =\; \backslash Z$ be the group of integers under addition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

, and let $H\; =\; 2\backslash Z$ be the subgroup consisting of the even integers. Then $2\backslash Z$ has two cosets in $\backslash Z$, namely the set of even integers and the set of odd integers, so the index $,\; \backslash Z:2\backslash Z,$ is 2. More generally, $,\; \backslash Z:n\backslash Z,\; =\; n$ for any positive integer ''n''.
When ''G'' is finite
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

, the formula may be written as $,\; G:H,\; =\; ,\; G,\; /,\; H,$, and it implies
Lagrange's theorem that $,\; H,$ divides $,\; G,$.
When ''G'' is infinite, $,\; G:H,$ is a nonzero cardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

that may be finite or infinite.
For example, $,\; \backslash Z:2\backslash Z,\; =\; 2$, but $,\; \backslash R:\backslash Z,$ is infinite.
If ''N'' is a normal subgroup
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

of ''G'', then $,\; G:N,$ is equal to the order of the quotient group
A quotient group or factor group is a math
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geome ...

$G/N$, since the underlying set of $G/N$ is the set of cosets of ''N'' in ''G''.
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 ahomomorphism
In algebra
Algebra (from ar, Ø§Ù„Ø¬Ø¨Ø±, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

, then the index of the kernel
Kernel may refer to:
Computing
* Kernel (operating system)
In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...

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 acting
Acting is an activity in which a story is told by means of its enactment
Enactment may refer to:
Law
* Enactment of a bill, when a bill becomes law
* Enacting formula, formulaic words in a bill or act which introduce its provisions
* Enactm ...

on a set ''X'', and let ''x'' âˆˆ ''X''. Then the cardinality
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). It ...

of the orbit
In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or po ...

of ''x'' under ''G'' is equal to the index of the stabilizer
Stabilizer, stabiliser, stabilisation or stabilization may refer to:
Chemistry and food processing
* Stabilizer (chemistry), a substance added to prevent unwanted change in state of another substance
** Polymer stabilizers are stabilizers used s ...

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 conjugates $gxg^$ of an element $x\; \backslash in\; G$ is equal to the index of the centralizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group (mathematics), group ''G'' is the set of elements C_G(S) of ''G'' such that each member g \in C_G(S) commutativity, commutes with ea ...

of ''x'' in ''G''.
* Similarly, the number of conjugates $gHg^$ of a subgroup ''H'' in ''G'' is equal to the index of the normalizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group (mathematics), group ''G'' is the set of elements C_G(S) of ''G'' such that each member g \in C_G(S) commutativity, commutes with ea ...

of ''H'' in ''G''.
* If ''H'' is a subgroup of ''G'', the index of the normal coreIn group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group (mathematics), group. The two most common types are the normal core of a subgroup and the p-core of a group.
The normal core Definition
For a grou ...

of ''H'' satisfies the following inequality:
::$,\; G:\backslash operatorname(H),\; \backslash le\; ,\; G:H,\; !$
:where ! denotes the factorial
In 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 g ...

function; this is discussed further below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Fred Below (1926â€“1988), American blues drummer
*Fritz von Below (1853â ...

.
:* 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
In 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 gen ...

of non-prime order, or more generally any perfect group
In mathematics, more specifically in group theory, a Group (mathematics), group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no trivial group, non-trivial abelian group, abelian quotient group, qu ...

.
Examples

* Thealternating group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

$A\_n$ has index 2 in the symmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

$S\_n,$ and thus is normal.
* The special orthogonal group
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). It ...

$\backslash operatorname(n)$ has index 2 in the orthogonal group
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). It ...

$\backslash operatorname(n)$, and thus is normal.
* The free abelian group
In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation that is associative, commutative, and invertible. A basis, also called ...

$\backslash Z\backslash oplus\; \backslash Z$ has three subgroups of index 2, namely
::$\backslash ,\backslash quad\; \backslash ,\backslash quad\backslash text\backslash quad\; \backslash $.
* More generally, if ''p'' is prime
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

then $\backslash Z^n$ has $(p^n-1)/(p-1)$ subgroups of index ''p'', corresponding to the $(p^n-1)$ nontrivial homomorphism
In algebra
Algebra (from ar, Ø§Ù„Ø¬Ø¨Ø±, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

s $\backslash Z^n\; \backslash to\; \backslash Z/p\backslash Z$.
* Similarly, the free group
for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the stud ...

$F\_n$ has $(p^n-1)$ subgroups of index ''p''.
* The infinite dihedral group
In mathematics, the infinite dihedral group Dihâˆž is an infinite group with properties analogous to those of the finite dihedral groups.
In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, ''p1m1'', se ...

has a 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 acardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

. For example, the index of ''H'' in ''G'' may be countable
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

or uncountable
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). It ...

, 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 anormal subgroup
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

''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 chiraloctahedral symmetry
A regular octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed ...

has 24 elements. It has a dihedral Dsymmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

Spyritohedral symmetry
150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection ...

also has 24 members and a subgroup of index 3 (this time it is a Dprismatic symmetry
In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dih''n'' ( ''n'' â‰¥ 2 ).
Types
There ar ...

group, see point groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group (mathematics), group o ...

), 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 mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

in the 6-member SNormal subgroups of prime power index

Normal subgroups ofprime power
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). It ...

index are kernels of surjective maps to ''p''-groups and have interesting structure, as described at Focal subgroup theorem: Subgroups and elaborated at focal subgroup theoremIn abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup
Peter Ludwig Mejdell Sylow () (12 December 1832 â€“ 7 September 1918) was a Norway, Norwegian mathematician who proved foundational results in gro ...

.
There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class:
* Eelementary abelian group
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). It ...

, and is the largest elementary abelian ''p''-group onto which ''G'' surjects.
* ASylow subgroup
Peter Ludwig Mejdell Sylow () (12 December 1832 â€“ 7 September 1918) was a Norway, Norwegian mathematician who proved foundational results in group theory. Biography
He was born and died in Oslo, Christiania (now Oslo). Sylow was a son of governm ...

s and the transfer homomorphism, as discussed there.
Geometric structure

An elementary observation is that one cannot have exactly 2 subgroups of index 2, as thecomplement
A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be:
* Complement (linguistics), a word or phrase having a particular syntactic role
** Subject complement, a word or phrase addi ...

of their symmetric difference
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). It ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

, 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
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). It ...

$\backslash mathbf\_p\; =\; \backslash mathbf/p.$ 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
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). It h ...

consisting of $p+1$ such subgroups.
For $p=2,$ the symmetric difference
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). It ...

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 â€“ it cannot contain exactly 2 or 4 index 2 subgroups, for instance.
See also

* Virtually *Codimension
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). It h ...

References

*External links

* *Subgroup of least prime index is normal

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{{DEFAULTSORT:Index Of A Subgroup Group theory