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{{short description|Mathematical group based upon a finite number of elements}}
{{Group theory sidebar |Basics}}

In [[abstract algebra]], a '''finite group''' is a [[group (mathematics)|group]] whose [[underlying set]] is [[finite set|finite]].  Finite groups often arise when considering symmetry of [[Symmetry in mathematics|mathematical]] or [[Symmetry (physics)|physical]] objects, when those objects admit just a finite number of structure-preserving transformations.  Important examples of finite groups include [[cyclic group]]s and [[permutation group]]s.

The study of finite groups has been an integral part of [[group theory]] since it arose in the 19th century.  One major area of study has been classification: the [[classification of finite simple groups]] (those with no nontrivial [[normal subgroup]]) was completed in 2004.

==History==

During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the [[Local analysis|local theory]] of finite groups and the theory of [[Solvable group|solvable]] and [[nilpotent group]]s.<ref>{{cite news | first = Michael | last = Aschbacher | author-link = Michael Aschbacher | year = 2004 | url = https://www.ams.org/notices/200407/fea-aschbacher.pdf | title = The Status of the Classification of the Finite Simple Groups | journal = [[Notices of the American Mathematical Society]] | volume = 51 | issue = 7 | pages = 736–740 }}</ref><ref>[[Daniel Gorenstein]] (1985), "The Enormous Theorem", ''Scientific American'', December 1, 1985, vol. 253, no. 6, pp.&nbsp;104–115.</ref>  As a consequence, the complete [[classification of finite simple groups]] was achieved, meaning that all those [[simple group]]s from which all finite groups can be built are now known.

During the second half of the twentieth century, mathematicians such as [[Claude Chevalley|Chevalley]] and [[Robert Steinberg|Steinberg]] also increased our understanding of finite analogs of [[classical groups]], and other related groups. One such family of groups is the family of [[general linear group]]s over [[finite field]]s.

Finite groups often occur when considering [[symmetry]] of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of [[Lie group]]s, which may be viewed as dealing with "[[continuous symmetry]]", is strongly influenced by the associated [[Weyl group]]s. These are finite groups generated by reflections which act on a finite-dimensional [[Euclidean space]]. The properties of finite groups can thus play a role in subjects such as [[theoretical physics]] and [[chemistry]].<ref>[https://chem.libretexts.org/Core/Physical_and_Theoretical_Chemistry/Group_Theory/Group_Theory_and_its_Application_to_Chemistry Group Theory and its Application to Chemistry] The Chemistry LibreTexts library</ref>

==Examples==

===Permutation groups===
{{Main|Permutation group}}
[[File:Symmetric group 4; Cayley graph 4,9.svg|thumb|320px|A [[Cayley graph]] of the symmetric group [[v:Symmetric group S4|S<sub>4</sub>]]]]
The '''symmetric group''' S<sub>''n''</sub> on a [[finite set]] of ''n'' symbols is the [[group (mathematics)|group]] whose elements are all the [[permutations]] of the ''n'' symbols, and whose [[group operation]] is the [[function composition|composition]] of such permutations, which are treated as [[bijection|bijective functions]] from the set of symbols to itself.<ref name=Jacobson-def>{{harvnb|Jacobson|2009|p=31}}</ref> Since there are ''n''! (''n'' [[factorial]]) possible permutations of a set of ''n'' symbols, it follows that the [[Order (group theory)|order]] (the number of elements) of the symmetric group S<sub>''n''</sub> is ''n''!.

===Cyclic groups===
{{Main|Cyclic group}}

A cyclic group Z<sub>''n''</sub> is a group all of whose elements are powers of a particular element ''a'' where {{nowrap|1=''a''{{i sup|''n''}} = ''a''{{i sup|0}} = e}}, the identity. A typical realization of this group is as the complex [[Root of unity|{{gaps|gap=0.12em|''n''|th}} roots of unity]]. Sending ''a'' to a [[primitive root of unity]] gives an isomorphism between the two. This can be done with any finite cyclic group.

===Finite abelian groups===
{{main|Finite abelian group}}
An '''[[abelian group]]''', also called a '''commutative group''', is a [[group (mathematics)|group]] in which the result of applying the group [[Operation (mathematics)|operation]] to two group elements does not depend on their order (the axiom of [[commutativity]]). They are named after [[Niels Henrik Abel]].<ref>{{harvnb|Jacobson|2009|p=41}}</ref>

An arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The [[automorphism group]] of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of [[Georg Frobenius]] and [[Ludwig Stickelberger]] and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of [[linear algebra]].

===Groups of Lie type===
{{Main|Group of Lie type}}
A '''group of Lie type'''  is a [[group (mathematics)|group]] closely related to the group  ''G''(''k'') of rational points of a reductive [[linear algebraic group]] ''G'' with values in the [[Field (mathematics)|field]] ''k''. Finite groups of Lie type give the bulk of nonabelian [[finite simple groups]].  Special cases include the [[classical groups]], the [[Chevalley groups]], the Steinberg groups, and the Suzuki–Ree groups.

Finite groups of Lie type were among the first groups to be considered in mathematics, after [[cyclic group|cyclic]], [[symmetric group|symmetric]] and [[alternating group|alternating]] groups, with the [[projective special linear group]]s over prime finite fields, PSL(2, ''p'') being constructed by [[Évariste Galois]] in the 1830s. The systematic exploration of finite groups of Lie type started with [[Camille Jordan]]'s theorem that the [[projective special linear group]] PSL(2, ''q'') is simple for ''q'' ≠ 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL(''n'', ''q'') of [[finite simple groups]]. Other classical groups were studied by [[Leonard Dickson]] in the beginning of 20th century. In the 1950s [[Claude Chevalley]] realized that after an appropriate reformulation, many theorems about [[semisimple Lie group]]s admit analogues for algebraic groups over an arbitrary field ''k'', leading to construction of what are now called ''Chevalley groups''. Moreover, as in the case of compact simple Lie groups, the corresponding groups turned out to be almost simple as abstract groups (''Tits simplicity theorem''). Although it was known since 19th century that other finite simple groups exist (for example, [[Mathieu groups]]), gradually a belief formed that nearly all finite simple groups can be accounted for by appropriate extensions of Chevalley's construction, together with cyclic and alternating groups. Moreover, the exceptions, the [[sporadic groups]], share many properties with the finite groups of Lie type, and in particular, can be constructed and characterized based on their ''geometry'' in the sense of Tits.

The belief has now become a theorem – the [[classification of finite simple groups]]. Inspection of the list of finite simple groups shows that groups of Lie type over a [[finite field]] include all the finite simple groups other than the cyclic groups, the alternating groups, the [[Tits group]], and the 26 [[sporadic simple group]]s.

==Main theorems==

===Lagrange's theorem===
{{Main|Lagrange's theorem (group theory)}}

For any finite group ''G'', the [[order (group theory)|order]] (number of elements) of every [[subgroup]] ''H'' of ''G'' divides the order of ''G''.  The theorem is named after [[Joseph-Louis Lagrange]].

===Sylow theorems===
{{Main|Sylow theorems}}

This provides a partial converse to Lagrange's theorem giving information about how many subgroups of a given order are contained in ''G''.

===Cayley's theorem===
{{main|Cayley's theorem}}
'''Cayley's theorem''', named in honour of [[Arthur Cayley]], states that every [[group (mathematics)|group]] ''G'' is [[group isomorphism|isomorphic]] to a [[subgroup]] of the [[symmetric group]] acting on ''G''.<ref>{{harvnb|Jacobson|2009|p=38}}</ref> This can be understood as an example of the [[Group action (mathematics)|group action]] of ''G'' on the elements of ''G''.<ref>{{harvnb|Jacobson|2009|p=72, ex. 1}}</ref>

===Burnside's theorem===
{{main|Burnside's theorem}}
'''Burnside's theorem''' in [[group theory]] states that if ''G'' is a finite group of [[Order (group theory)|order]] ''p''{{sup|''a''}}''q''{{sup|''b''}}, where ''p'' and ''q'' are [[prime number]]s, and ''a'' and ''b'' are [[negative and positive numbers|non-negative]] [[integer]]s, then ''G'' is [[Solvable group|solvable]]. Hence each
non-Abelian [[finite simple group]] has order divisible by at least three distinct primes.

===Feit–Thompson theorem===
The '''Feit–Thompson theorem''', or '''odd order theorem''', states that every finite [[group (mathematics)|group]] of odd [[order (group theory)|order]] is [[solvable group|solvable]]. It was proved by {{harvs|last=Feit|first=Walter|authorlink=Walter Feit|first2=John Griggs |last2=Thompson|author2-link=John Griggs Thompson|year1=1962|year2=1963|txt=yes}}

===Classification of finite simple groups===
The [[classification of finite simple groups]] is a theorem stating that every [[List of finite simple groups|finite simple group]] belongs to one of the following families:
* A [[cyclic group]] with prime order;
* An [[alternating group]] of degree at least 5;
* A [[Group of Lie type|simple group of Lie type]];
* One of the 26 [[sporadic group|sporadic simple group]]s;
* The [[Tits group]] (sometimes considered as a 27th sporadic group).

The finite simple groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the [[prime number]]s are the basic building blocks of the [[natural number]]s.  The [[Jordan–Hölder theorem]] is a more precise way of stating this fact about finite groups. However, a significant difference with respect to the case of [[integer factorization]] is that such "building blocks" do not necessarily determine uniquely a group, since there might be many non-isomorphic groups with the same [[composition series]] or, put in another way, the [[group extension#Extension problem|extension problem]] does not have a unique solution.

The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. [[Daniel Gorenstein|Gorenstein]] (d.1992), [[Richard Lyons (mathematician)|Lyons]], and [[Ronald Solomon|Solomon]] are gradually publishing a simplified and revised version of the proof.

==Number of groups of a given order==
Given a positive integer ''n'', it is not at all a routine matter to determine how many [[isomorphism]] types of groups of [[group order|order]] ''n'' there are. Every group of [[prime number|prime]]  order is [[cyclic group|cyclic]], because [[Lagrange's theorem (group theory)|Lagrange's theorem]] implies that the cyclic subgroup generated by any of its non-identity elements is the whole group.
If ''n'' is the square of a prime, then there are exactly two possible isomorphism types of group of order ''n'', both of which are abelian. If ''n'' is a higher power of a prime, then results of [[Graham Higman]] and [[Charles Sims (mathematician)|Charles Sims]] give asymptotically correct estimates for the number of isomorphism types of groups of order ''n'', and the number grows very rapidly as the power increases.

Depending on the prime factorization of ''n'', some restrictions may be placed on the structure of groups of order ''n'', as a consequence, for example, of results such as the [[Sylow theorems]]. For example, every group of order ''pq'' is cyclic when {{nowrap|''q'' < ''p''}} are primes with {{nowrap|''p'' − 1}} not divisible by ''q''. For a necessary and sufficient condition, see [[cyclic number (group theory)|cyclic number]].

If ''n'' is [[Square-free integer|squarefree]], then any group of order ''n'' is solvable. [[Burnside's theorem]], proved using [[Character theory|group characters]], states that every group of order ''n'' is solvable when ''n'' is divisible by fewer than three distinct primes, i.e. if {{nowrap|1=''n'' = ''p''<sup>''a''</sup>''q''<sup>''b''</sup>}}, where ''p'' and ''q'' are prime numbers, and ''a'' and ''b'' are non-negative integers. By the [[Feit–Thompson theorem]], which has a long and complicated proof, every group of order ''n'' is solvable when ''n'' is odd.

For every positive integer ''n'', most groups of order ''n'' are [[Solvable group|solvable]]. To see this for any particular order is usually not difficult (for example, there is, up to isomorphism,  one non-solvable group and 12 solvable groups of order 60) but the proof of this for all orders uses the [[classification of finite simple groups]]. For any positive integer ''n'' there are at most two simple groups of order ''n'', and there are infinitely many positive integers ''n'' for which there are two non-isomorphic simple groups of order ''n''.

===Table of distinct groups of order ''n''===
{{main|oeis:A000001|oeis:A000688|oeis:A060689}}

{| class="wikitable"
|-----
! Order ''n''
! # Groups<ref>{{cite book |first=John F. |last=Humphreys |title=A Course in Group Theory |publisher=Oxford University Press |year=1996 |isbn=0198534590 |pages=238–242 |zbl=0843.20001}}</ref>
! Abelian
! Non-Abelian
|-----
! 0
| 0
| 0
| 0
|-----
! 1
| 1
| 1
| 0
|-----
! 2
| 1
| 1
| 0
|-----
! 3
| 1
| 1
| 0
|-----
! 4
| 2
| 2
| 0
|-----
! 5
| 1
| 1
| 0
|-----
! 6
| 2
| 1
| 1
|-----
! 7
| 1
| 1
| 0
|-----
! 8
| 5
| 3
| 2
|-----
! 9
| 2
| 2
| 0
|-----
! 10
| 2
| 1
| 1
|-----
! 11
| 1
| 1
| 0
|-----
! 12
| 5
| 2
| 3
|-----
! 13
| 1
| 1
| 0
|-----
! 14
| 2
| 1
| 1
|-----
! 15
| 1
| 1
| 0
|-----
! 16
| 14
| 5
| 9
|-----
! 17
| 1
| 1
| 0
|-----
! 18
| 5
| 2
| 3
|-----
! 19
| 1
| 1
| 0
|-----
! 20
| 5
| 2
| 3
|-----
! 21
| 2
| 1
| 1
|-----
! 22
| 2
| 1
| 1
|-----
! 23
| 1
| 1
| 0
|-----
! 24
| 15
| 3
| 12
|-----
! 25
| 2
| 2
| 0
|-----
! 26
| 2
| 1
| 1
|-----
! 27
| 5
| 3
| 2
|-----
! 28
| 4
| 2
| 2
|-----
! 29
| 1
| 1
| 0
|-----
! 30
| 4
| 1
| 3
|}

==See also==
{{Div col|colwidth=30em}}
*[[Association scheme]]
*[[Cauchy's theorem (group theory)]]
*[[Classification of finite simple groups]]
*[[Commuting probability]]
*[[Finite ring]]
*[[Finite-state machine]]

*[[Infinite group]]

*[[List of finite simple groups]]
*[[List of small groups]]
*[[Modular representation theory]]
*[[Monstrous moonshine]]
*[[P-group]]
*[[Profinite group]]
*[[Representation theory of finite groups]]

{{div col end}}
==References==
{{Reflist}}

==Further reading==
{{refbegin}}
* {{Cite book| last=Jacobson| first=Nathan | year=2009| title=Basic Algebra I | edition=2nd | publisher=[[Dover Publications]] | isbn = 978-0-486-47189-1 }}
{{refend}}

==External links==
* {{OEIS el|1=A000001|2=Number of groups of order n}}
* {{OEIS el|1=A000688|2=Number of Abelian groups of order ''n''|formalname=Number of Abelian groups of order n; number of factorizations of n into prime powers}}
* {{OEIS el|1=A060689|2=Number of non-Abelian groups of order n}}
* Small groups on [http://groupnames.org GroupNames]
* A [http://www.bluetulip.org/programs/finitegroups.html classifier] for groups of small order

{{Authority control}}

[[Category:Finite groups| ]]
[[Category:Properties of groups]]