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A group with finitely many elements. The number of elements is called the order of the group. Historically, many concepts in abstract group theory have had their origin in the theory of finite groups.

It is usually said that the aim of finite group theory is to describe the groups of given order up to isomorphism. This is only partly true. A naïve approach based on a wish to classify all groups is doomed to failure. For example, the compilation of a list of all non-isomorphic groups of the comparatively small order 1024 would constitute a difficult task for the best modern computers. In general, the classification of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f0402702.png" />-groups (groups of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f0402703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f0402704.png" /> a prime number) is a  "wild"  problem. On the other hand, there do exist some extremal classes of groups, playing a central role in the theory, for which the classification problem (enumeration up to isomorphism) is either solved or is a perfectly reasonable objective.

Thus there is a complete theory of finite Abelian groups (see [[Abelian group|Abelian group]]) and various classes of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f0402705.png" />-groups (regular, free of exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f0402706.png" />, of maximal class, those given by generators and the least possible number of relations) allow of a qualitative description. The description of finite groups of integral invertible matrices is an example of an important classification problem; it has been solved only for small orders of the matrices.

During the formative period in the theory of finite groups, which is associated with the names of A.L. Cauchy, J.L. Lagrange, C.F. Gauss, N.H. Abel, and later E. Galois, finite groups were studied almost entirely in the form of permutation groups (whence the unsuccessful well-established term substitution group, cf. [[Permutation group|Permutation group]]), and this point of view is still fruitful today. For example, if an abstract finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f0402707.png" /> can be realized as a multiply-transitive permutation group (that is, as a suitable subgroup of a symmetric group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f0402708.png" />), then in a number of cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f0402709.png" /> is uniquely determined up to isomorphism. Classification results for simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027010.png" />-transitive groups constitute a rich section of finite group theory, whose foundations were laid more than a hundred years ago by C. Jordan. The computational methods in the theory of permutation groups, developed in recent years, have been called into existence partly by the requirements of combinatorics, graph theory, coding theory, and the need to check various conjectures.

The theory of linear representations of finite groups, developed at the beginning of the 20th century by G. Frobenius, W. Burnside, I. Schur, and others (see [[Finite group, representation of a|Finite group, representation of a]]), provided a powerful tool for investigating abstract groups, and served as a prototype for the analogous representation theory of Lie groups, and later of other algebraic systems. The formulation of properties of groups and their representations in terms of the theory of characters is one manifestation of the mutual relationships between the various branches of algebra. Representations of finite groups over commutative rings and fields of non-zero characteristic, intensively investigated by R. Brauer and others (see [[#References|[2]]]), have been exploited to yield many deep results.

Extensions of finite groups are studied using the cohomology theory of groups (see [[Cohomology of groups|Cohomology of groups]]), which is also of independent interest.

The almost obvious idea that properties of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027011.png" /> must to some extent be arithmetical and depend on the canonical prime factorization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027012.png" /> of its order, is given precise form in the [[Sylow theorems|Sylow theorems]] on the existence and conjugacy of subgroups of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027016.png" /> are relatively prime natural numbers, then a subgroup of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027017.png" />, called a Hall subgroup, need not, in general, exist. However, for the large class of solvable groups, Hall subgroups exist and fully inherit the properties of Sylow subgroups. The solvability of groups of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027018.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027020.png" /> are prime numbers), discovered by Burnside, is a typical example of an arithmetical property. The solvability of groups of odd order, established many years later by J. Thompson and W. Feit (see [[Burnside problem|Burnside problem]], 1), is an achievement which has given finite group theory a radically new orientation. The existence of involutions (elements of order 2) in non-solvable groups — an alternative formulation of the Feit–Thompson theorem — and the expression of properties of simple groups in terms of centralizers of involutions have formed one of the principal directions of research in finite group theory (the realization of Brauer's programme).

In the roaring period 1961–1981, marked by constant activity of algebraists in various countries, one of the most remarkable achievements in the history of mathematics was made: the classification of the finite simple groups (cf. [[Simple finite group|Simple finite group]]). Regrettably, a drawback of this classification is the fact that until now it has not been verified, because of the absence of one integral text and the exceptional complexity of the structure of the reasoning used. Therefore the task of rendering individual parts of the proof more transparent has not faded away, and the branch of the theory occupied with the finite simple groups continues to be an active one. All kinds of consequences of the classification are of interest (e.g., the description of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027021.png" />-transitive groups), although many mathematicians call results of this kind conditional, out of precaution.

Parallel with the classification direction, an intensive study was made of invariants of the finite linear groups and of properties of representations. Especially fruitful has been the work concerning the description of all complex characters of the finite simple groups of Lie type. Substantial progress was made in this case, a symbol of this being the theory of Deligne–Lusztig, which was brought about by combining methods of algebraic geometry and the theory of algebraic groups. Nowadays, this theory is one of the most splendid parts of algebra, on the interface between the theory of finite groups and its applications in most varied branches of mathematics.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Klein,   "Development of mathematics in the 19th century" , Math. Sci. Press  (1979)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.W. Curtis,   I. Reiner,   "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.I. Kargapolov,   J.I. [Yu.I. Merzlyakov] Merzljakov,   "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.I. Kostrikin,   "Finite groups"  ''Itogi Nauk. Algebra 1964''  (1966)  pp. 7–46  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.A. Chunikhin,   L.A. Shemetkov,   "Finite groups"  ''Itogi Nauk. Algebra. Topol. Geom. 1969''  (1971)  pp. 7–70  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.D. Mazurov,   "Finite groups"  ''Itogi Nauk. Algebra. Topol. Geom.'' , '''14'''  (1976)  pp. 5–56  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A. Speiser,   "Die Theorie der Gruppen von endlicher Ordnung" , Birkhäuser  (1956)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  H. Wielandt,   "Finite permutation groups" , Acad. Press  (1968)  (Translated from German)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  B. Huppert,   "Endliche Gruppen" , '''1''' , Springer  (1967)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  D. Gorenstein,   "Finite groups" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  I.M. Isaacs,   "Character theory of finite groups" , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  M. Aschbacher,   "Finite group theory" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  D. Gorenstein,   "Finite simple groups. An introduction to their classification" , Plenum  (1982)</TD></TR></table>



====Comments====
In order to appreciate the difficulty of a naïve approach, cf. the classification of the 267 groups of order 64 in [[#References|[a1]]].

The finite simple groups have been classified only recently (as mentioned in the article above; they include the Chevalley groups (plus twisted types), the alternating groups of degrees at least 5 and the 26 so-called sporadic groups; for details see [[Simple finite group|Simple finite group]]). See [[#References|[13]]] for a detailed assessment as to 1982. As a consequence all permutation groups which are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027022.png" />-transitive or primitive of rank 3 (i.e., have 3 orbits on the set of all unordered pairs) have been determined, cf. [[#References|[a5]]]. For further consequences see [[#References|[a6]]], [[#References|[a2]]].

Burnside's result on the solvability of groups of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027023.png" /> has been recently proved by D.M. Goldschmidt and H. Matsuyama without using character theory (cf. [[Character of a group|Character of a group]], and [[#References|[a4]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hall,   "The groups of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027024.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040270/f04027025.png" />)" , Macmillan  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.M. Kantor,   "Some consequences of the classification of finite simple groups"  ''Contempory Math.'' , '''45'''  (1985)  pp. 159–173</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Blackburn,   B. Huppert,   "Finite groups" , '''1–2''' , Springer  (1981–1982)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Suzuki,   "Group theory" , '''1–2''' , Springer  (1986)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.W. Liebeck,   "The affine permutation groups of rank three"  ''Proc. London Math. Soc. (3)'' , '''54'''  (1987)  pp. 477–516</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  W. Feit,   "Some consequences of the classification of finite simple groups" , ''Proc. Symp. Pure Math.'' , '''37''' , Amer. Math. Soc.  (1980)  pp. 175–181</TD></TR></table>