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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 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 ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. 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 :H/math> or (G:H). Because ''G'' is the disjoint union of the left cosets and because each left coset has the same
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 = \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 ...

addition
, and let H = 2\Z be the subgroup consisting of the even integers. Then 2\Z has two cosets in \Z, namely the set of even integers and the set of odd integers, so the index , \Z:2\Z, is 2. More generally, , \Z:n\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, , \Z:2\Z, = 2, but , \R:\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, \,, H:K, . * If ''H'' and ''K'' are subgroups of ''G'', then ::, G:H\cap K, \le , G : H, \,, G : K, , :with equality if HK=G. (If , G:H\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\cap K, \le , G:K, , :with equality if HK=G. (If , H:H\cap K, is finite, then equality holds if and only if HK=G.) * If ''G'' and ''H'' are groups and \varphi \colon G\to H is a
homomorphism In algebra Algebra (from ar, الجبر, 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 \varphi in ''G'' is equal to the order of the image: ::, G:\operatorname\;\varphi, =, \operatorname\;\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'' ∈ ''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, .\! :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 \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:\operatorname(H), \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 (1926–1988), American blues drummer *Fritz von Below (1853 ...
. :* 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

* The
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 ...
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 ...
\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 ...
\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 ...
\Z\oplus \Z has three subgroups of index 2, namely ::\,\quad \,\quad\text\quad \. * 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 \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, الجبر, 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 \Z^n \to \Z/p\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 Dih∞ 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 ...

infinite dihedral group
has a
cyclic subgroup
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 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 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), ...
''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 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 D4 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 D2 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 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 ...
S3. All elements from any particular coset of ''A'' perform the same permutation of the cosets of ''H''. On the other hand, the group Th of
pyritohedral 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 D2h
prismatic 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'' ≥ 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 S3 symmetric group.


Normal subgroups of prime power index

Normal subgroups of
prime 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 – 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: * E''p''(''G'') is the intersection of all index ''p'' normal subgroups; ''G''/E''p''(''G'') is an
elementary 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. * 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/math>): ''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 :\mathbf^p(G) \supseteq \mathbf^p(G) \supseteq \mathbf^p(G). These groups have important connections to the
Sylow subgroup Peter Ludwig Mejdell Sylow () (12 December 1832 – 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 the
complement 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/\mathbf^p(G) \cong (\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 :\mathbf(\operatorname(G,\mathbf/p)). In detail, the space of homomorphisms from ''G'' to the (cyclic) group of order ''p,'' \operatorname(G,\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 ...
\mathbf_p = \mathbf/p. A non-trivial such map has as kernel a normal subgroup of index ''p,'' and multiplying the map by an element of (\mathbf/p)^\times (a non-zero number mod ''p'') does not change the kernel; thus one obtains a map from :\mathbf(\operatorname(G,\mathbf/p)) := (\operatorname(G,\mathbf/p))\setminus\)/(\mathbf/p)^\times to normal index ''p'' subgroups. Conversely, a normal subgroup of index ''p'' determines a non-trivial map to \mathbf/p up to a choice of "which coset maps to 1 \in \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+\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,\ldots index 2 subgroups – 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
a
Groupprops, The Group Properties Wiki
{{DEFAULTSORT:Index Of A Subgroup Group theory