HOME

TheInfoList



OR:

The following list in mathematics contains the finite groups of small
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
group isomorphism In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two g ...
.


Counts

For ''n'' = 1, 2, … the number of nonisomorphic groups of order ''n'' is : 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, ... For labeled groups, see .


Glossary

Each group is named by their Small Groups library as G''o''''i'', where ''o'' is the order of the group, and ''i'' is the index of the group within that order. Common group names: * Z''n'': the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...
of order ''n'' (the notation C''n'' is also used; it is isomorphic to the
additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structure ...
of Z/''n''Z). * Dih''n'': the
dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
of order 2''n'' (often the notation D''n'' or D2''n'' is used ) ** K4: the
Klein four-group In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third on ...
of order 4, same as and Dih2. * S''n'': the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
of degree ''n'', containing the ''n''!
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...
s of ''n'' elements. * A''n'': the
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...
of degree ''n'', containing the
even permutation In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ...
s of ''n'' elements, of order 1 for , and order ''n''!/2 otherwise. * Dic''n'' or Q4''n'': the dicyclic group of order 4''n''. ** Q8: the
quaternion group In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a nonabelian group, non-abelian group (mathematics), group of Group order, order eight, isomorphic to the eight-element subset \ of the quaternions under multiplication. ...
of order 8, also Dic2. The notations Z''n'' and Dih''n'' have the advantage that
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 of all isometries t ...
C''n'' and D''n'' do not have the same notation. There are more isometry groups than these two, of the same abstract group type. The notation denotes the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of the two groups; ''G''''n'' denotes the direct product of a group with itself ''n'' times. ''G'' ⋊ ''H'' denotes a
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in w ...
where ''H''
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
on ''G''; this may also depend on the choice of action of ''H'' on ''G''.
Abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
and
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The da ...
s are noted. (For groups of order , the simple groups are precisely the cyclic groups Z''n'', for
prime A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only way ...
''n''.) The equality sign ("=") denotes isomorphism. The
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
in the cycle graphs is represented by the black circle. The lowest order for which the cycle graph does not uniquely represent a group is order 16. In the lists of
subgroup In group theory, a branch of mathematics, given a 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'' is a subgrou ...
s, the trivial group and the group itself are not listed. Where there are several isomorphic subgroups, the number of such subgroups is indicated in parentheses.
Angle brackets A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
show the
presentation of a group In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—an ...
.


List of small abelian groups

The finite abelian groups are either cyclic groups, or direct products thereof; see
Abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
. The numbers of nonisomorphic abelian groups of orders ''n'' = 1, 2, ... are : 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, ... For labeled abelian groups, see .


List of small non-abelian groups

The numbers of non-abelian groups, by order, are counted by . However, many orders have no non-abelian groups. The orders for which a non-abelian group exists are : 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, ...


Classifying groups of small order

Small groups of
prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, ...
order ''p''''n'' are given as follows: *Order ''p'': The only group is cyclic. *Order ''p''2: There are just two groups, both abelian. *Order ''p''3: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order ''p''2 by a cyclic group of order ''p''. The other is the quaternion group for and a group of exponent ''p'' for . *Order ''p''4: The classification is complicated, and gets much harder as the exponent of ''p'' increases. Most groups of small order have a Sylow ''p'' subgroup ''P'' with a normal ''p''-complement ''N'' for some prime ''p'' dividing the order, so can be classified in terms of the possible primes ''p'', ''p''-groups ''P'', groups ''N'', and actions of ''P'' on ''N''. In some sense this reduces the classification of these groups to the classification of ''p''-groups. Some of the small groups that do not have a normal ''p''-complement include: *Order 24: The symmetric group S4 *Order 48: The binary octahedral group and the product *Order 60: The alternating group A5. The smallest order for which it is ''not'' known how many nonisomorphic groups there are is 2048 = 211.


Small Groups Library

The GAP
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The ...
contains a package called the "Small Groups library," which provides access to descriptions of small order groups. The groups are listed
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. At present, the library contains the following groups:Hans Ulrich Besch
The Small Groups library
* those of order at most 2000 except for order 1024 ( groups in the library; the ones of order 1024 had to be skipped, as there are an additional nonisomorphic 2-groups of order 1024); * those of cubefree order at most 50000 (395 703 groups); * those of squarefree order; * those of order ''p''''n'' for ''n'' at most 6 and ''p'' prime; * those of order ''p''7 for ''p'' = 3, 5, 7, 11 (907 489 groups); * those of order ''pq''''n'' where ''q''''n'' divides 28, 36, 55 or 74 and ''p'' is an arbitrary prime which differs from ''q''; * those whose orders factorise into at most 3 primes (not necessarily distinct). It contains explicit descriptions of the available groups in computer readable format. The smallest order for which the SmallGroups library does not have information is 1024.


See also

*
Classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or els ...
*
Composition series In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natu ...
* List of finite simple groups * Number of groups of a given order *
Small Latin squares and quasigroups Latin squares and quasigroups are equivalent mathematical objects, although the former has a combinatorial nature while the latter is more algebraic. The listing below will consider the examples of some very small ''orders'', which is the side len ...
*
Sylow theorems In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixe ...


Notes


References

*, Table 1, Nonabelian groups order<32. *


External links

* Particular groups in the Group Properties Wiki * Groups of given order * {{cite web , first1 = H. U. , last1 = Besche , last2 = Eick , first2 = B. , last3 = O'Brien , first3 = E. , url = http://www.icm.tu-bs.de/ag_algebra/software/small/ , title = small group library , url-status = dead , archive-url = https://web.archive.org/web/20120305020857/http://www.icm.tu-bs.de/ag_algebra/software/small/ , archive-date = 2012-03-05
GroupNames database
Groups that are small Groups that are small Finite groups