Editing Group theory (section)

===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''.