Editing Group theory (section)

==Main classes of groups==
{{Main|Group (mathematics)}}

The range of groups being considered has gradually expanded from finite permutation groups and special examples of [[matrix group]]s to abstract groups that may be specified through a [[presentation of a group|presentation]] by [[Generating set of a group|generators]] and [[Binary relation|relations]].

===Permutation groups===
The first [[Class (set theory)|class]] of groups to undergo a systematic study was [[permutation group]]s. Given any set ''X'' and a collection ''G'' of [[bijection]]s of ''X'' into itself (known as ''permutations'') that is closed under compositions and inverses, ''G'' is a group [[Group action (mathematics)|acting]] on ''X''. If ''X'' consists of ''n'' elements and ''G'' consists of ''all'' permutations, ''G'' is the [[symmetric group]] S<sub>''n''</sub>; in general, any permutation group ''G'' is a [[subgroup]] of the symmetric group of ''X''. An early construction due to [[Arthur Cayley|Cayley]] exhibited any group as a permutation group, acting on itself ({{nowrap|1=''X'' = ''G''}}) by means of the left [[regular representation]].

In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for {{nowrap|''n'' ≥ 5}}, the [[alternating group]] A<sub>''n''</sub> is [[simple group|simple]], i.e. does not admit any proper [[normal subgroup]]s. This fact plays a key role in the [[Abel–Ruffini theorem|impossibility of solving a general algebraic equation of degree {{nowrap|''n'' ≥ 5}} in radicals]].

===Matrix groups===
The next important class of groups is given by ''matrix groups'', or [[linear group]]s. Here ''G'' is a set consisting of invertible [[matrix (mathematics)|matrices]] of given order ''n'' over a [[field (mathematics)|field]] ''K'' that is closed under the products and inverses. Such a group acts on the ''n''-dimensional vector space ''K''<sup>''n''</sup> by [[linear transformation]]s. This action makes matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group ''G''.

===Transformation groups===
Permutation groups and matrix groups are special cases of [[transformation group]]s: groups that act on a certain space ''X'' preserving its inherent structure. In the case of permutation groups, ''X'' is a set; for matrix groups, ''X'' is a [[vector space]]. The concept of a transformation group is closely related with the concept of a [[symmetry group]]: transformation groups frequently consist of ''all'' transformations that preserve a certain structure.
  
The theory of transformation groups forms a bridge connecting group theory with [[differential geometry]]. A long line of research, originating with [[Sophus Lie|Lie]] and [[Felix Klein|Klein]], considers group actions on [[manifold]]s by [[homeomorphism]]s or [[diffeomorphism]]s. The groups themselves may be [[discrete group|discrete]] or [[continuous group|continuous]].

===Abstract groups===
Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an '''abstract group''' began to take hold, where "abstract" means that the nature of the elements are ignored in such a way that two [[group isomorphism|isomorphic groups]] are considered as the same group. A typical way of specifying an abstract group is through a [[presentation of a group|presentation]] by ''generators and relations'',

: <math> G = \langle S|R\rangle. </math>

A significant source of abstract groups is given by the construction of a ''factor group'', or [[quotient group]], ''G''/''H'', of a group ''G'' by a [[normal subgroup]] ''H''. [[Class group]]s of [[algebraic number field]]s were among the earliest examples of factor groups, of much interest in [[number theory]]. If a group ''G'' is a permutation group on a set ''X'', the factor group ''G''/''H'' is no longer acting on ''X''; but the idea of an abstract group permits one not to worry about this discrepancy.

The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under [[isomorphism]], as well as the classes of group with a given such property: [[finite group]]s, [[periodic group]]s, [[simple group]]s, [[solvable group]]s, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of [[abstract algebra]] in the works of [[David Hilbert|Hilbert]], [[Emil Artin]], [[Emmy Noether]], and mathematicians of their school.{{citation needed|date=June 2012}}

=== Groups with additional structure ===
An important elaboration of the concept of a group occurs if ''G'' is endowed with additional structure, notably, of a [[topological space]], [[differentiable manifold]], or [[algebraic variety]]. If the multiplication and inversion of the group are compatible with this structure, that is, they are [[continuous map|continuous]], [[smooth map|smooth]] or [[Regular map (algebraic geometry)|regular]] (in the sense of algebraic geometry) maps, then ''G'' is a [[topological group]], a [[Lie group]], or an [[algebraic group]].<ref>This process of imposing extra structure has been formalized through the notion of a  [[group object]] in a suitable [[category (mathematics)|category]]. Thus Lie groups are group objects in the category of differentiable manifolds and affine algebraic groups are group objects in the category of affine algebraic varieties.</ref>

The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for [[abstract harmonic analysis]], whereas [[Lie group]]s (frequently realized as transformation groups) are the mainstays of [[differential geometry]] and unitary [[representation theory]]. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, [[compact Lie group|compact connected Lie groups]] have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group ''Γ'' can be realized as a [[lattice (discrete subgroup)|lattice]] in a topological group ''G'', the geometry and analysis pertaining to ''G'' yield important results about ''Γ''. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups ([[profinite group]]s): for example, a single [[powerful p-group|''p''-adic analytic group]] ''G'' has a family of quotients which are finite [[p-group|''p''-groups]] of various orders, and properties of ''G'' translate into the properties of its finite quotients.