Editing Group theory (section)

==Branches of group theory==

===Finite group theory===
{{Main|Finite group}}
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.{{citation needed|date=December 2013|reason=In who's opinion?}} 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 group]]s, 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]].

===Representation of groups===
{{Main|Representation theory}}
Saying that a group ''G'' ''[[Group action (mathematics)|acts]]'' on a set ''X'' means that every element of ''G'' defines a bijective map on the set ''X'' in a way compatible with the group structure. When ''X'' has more structure, it is useful to restrict this notion further: a representation of ''G'' on a [[vector space]] ''V'' is a [[group homomorphism]]:

:<math>\rho:G \to \operatorname{GL}(V),</math>

where [[general linear group|GL]](''V'') consists of the invertible [[linear map|linear transformations]] of ''V''. In other words, to every group element ''g'' is assigned an [[automorphism]] ''ρ''(''g'') such that {{nowrap|1=''ρ''(''g'') ∘ ''ρ''(''h'') = ''ρ''(''gh'')}} for any ''h'' in ''G''.

This definition can be understood in two directions, both of which give rise to whole new domains of mathematics.<ref>Such as [[group cohomology]] or [[Equivariant algebraic K-theory|equivariant K-theory]].</ref> On the one hand, it may yield new information about the group ''G'': often, the group operation in ''G'' is abstractly given, but via ''ρ'', it corresponds to the [[matrix multiplication|multiplication of matrices]], which is very explicit.<ref>In particular, if the representation is [[faithful representation|faithful]].</ref> On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if ''G'' is finite, it is known that ''V'' above decomposes into [[irreducible representation|irreducible parts]] (see [[Maschke's theorem]]). These parts, in turn, are much more easily manageable than the whole ''V'' (via [[Schur's lemma]]).

Given a group ''G'', [[representation theory]] then asks what representations of ''G'' exist. There are several settings, and the employed methods and obtained results are rather different in every case: [[representation theory of finite groups]] and representations of [[Lie group]]s are two main subdomains of the theory. The totality of representations is governed by the group's [[character theory|characters]]. For example, [[Fourier series|Fourier polynomial]]s can be interpreted as the characters of [[unitary group|U(1)]], the group of [[complex numbers]] of [[absolute value]] ''1'', acting on the [[Lp space|''L''<sup>2</sup>]]-space of periodic functions.

===Lie theory===
{{Main|Lie theory}}
A [[Lie group]] is a [[group (mathematics)|group]] that is also a [[differentiable manifold]], with the property that the group operations are compatible with the [[Differential structure|smooth structure]]. Lie groups are named after [[Sophus Lie]], who laid the foundations of the theory of continuous [[transformation group]]s. The term ''groupes de Lie'' first appeared in French in 1893 in the thesis of Lie's student [[:pt:Arthur Tresse|Arthur Tresse]], page 3.<ref>{{citation |title= Sur les invariants différentiels des groupes continus de transformations | author= Arthur Tresse |journal=Acta Mathematica|volume=18|year=1893|pages=1–88 |doi=10.1007/bf02418270|url=https://zenodo.org/record/2273334|doi-access=free}}</ref>

Lie groups represent the best-developed theory of [[continuous symmetry]] of [[mathematical object]]s and [[mathematical structure|structures]], which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern [[theoretical physics]]. They provide a natural framework for analysing the continuous symmetries of [[differential equations]] ([[differential Galois theory]]), in much the same way as permutation groups are used in [[Galois theory]] for analysing the discrete symmetries of [[algebraic equations]]. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.

===Combinatorial and geometric group theory===
{{Main|Geometric group theory}}
Groups can be described in different ways. Finite groups can be described by writing down the [[group table]] consisting of all possible multiplications {{nowrap|''g'' • ''h''}}. A more compact way of defining a group is by ''generators and relations'', also called the ''presentation'' of a group. Given any set ''F'' of generators <math>\{g_i\}_{i\in I}</math>, the [[free group]] generated by ''F'' surjects onto the group ''G''. The kernel of this map is called the subgroup of relations, generated by some subset ''D''. The presentation is usually denoted by <math>\langle F \mid D\rangle.</math> For example, the group presentation <math>\langle a,b\mid aba^{-1}b^{-1}\rangle</math> describes a group which is isomorphic to <math>\mathbb{Z}\times\mathbb{Z}.</math> A string consisting of generator symbols and their inverses is called a ''word''.

[[Combinatorial group theory]] studies groups from the perspective of generators and relations.<ref>{{harvnb|Schupp|Lyndon|2001}}</ref> It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of [[graph (discrete mathematics)|graph]]s via their [[fundamental group]]s. A fundamental theorem of this area is that every subgroup of a free group is free.

There are several natural questions arising from giving a group by its presentation. The ''[[word problem for groups|word problem]]'' asks whether two words are effectively the same group element. By relating the problem to [[Turing machine]]s, one can show that there is in general no [[algorithm]] solving this task. Another, generally harder, algorithmically insoluble problem is the [[group isomorphism problem]], which asks whether two groups given by different presentations are actually isomorphic. For example, the group with presentation <math>\langle x,y \mid xyxyx = e \rangle,</math> is isomorphic to the additive group '''Z''' of integers, although this may not be immediately apparent. (Writing <math>z=xy</math>, one has <math>G \cong \langle z,y \mid z^3 = y\rangle \cong \langle z\rangle.</math>)

[[File:Cayley graph of F2.svg|right|150px|thumb|The Cayley graph of &lang; x, y ∣ &rang;, the free group of rank 2]]
[[Geometric group theory]] attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on.<ref>{{harvnb|La Harpe|2000}}</ref> The first idea is made precise by means of the [[Cayley graph]], whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the [[word metric]] given by the length of the minimal path between the elements. A theorem of [[John Milnor|Milnor]] and Svarc then says that given a group ''G'' acting in a reasonable manner on a [[metric space]] ''X'', for example a [[compact manifold]], then ''G'' is [[quasi-isometry|quasi-isometric]] (i.e. looks similar from a distance) to the space ''X''.