Infinite group

In group theory, an area of mathematics, an infinite group is a group whose underlying set contains infinitely many elements. In other words, it is a group of infinite order. The structure of infinite groups is often a question of mathematical analysis of the asymptotics of how various invariants grow relative to a generating set, or how a group acts on a topological or measure space. In contrast, the structure of finite groups is determined largely by methods of abstract algebra.

Examples

(Z, +), the group of integers with addition is infinite, as is (Q, +) the group of rational numbers with addition
Non-discrete Lie groups are infinite. For example, (R, +), the group of real numbers with addition is an infinite group
The general linear group of order n > 0 over an infinite field is infinite
The Infinite dihedral group
The Tarski monster group
The Prüfer p-group
A free group
SL(2,Z)
Amenable groups
Thompson groups
Grigorchuk group
Lamplighter groups

Torsion

An infinite group is called a torsion group if every element has finite order. Examples include the Prüfer p-group and certain Burnside groups. In contrast, a group is torsion-free if no non-identity element has finite order, such as ℤ or free groups.

Asymptotics

An infinite group equipped with a generating set $\{\gamma _{i}\mid i\in I\}$ inherits a natural metric structure via the word metric. This is the unique left-invariant distance function such that $d(\gamma _{i},e)=d(\gamma _{i}^{-1},e)=1$ for each generator $\gamma _{i}$ , and distances extend by minimal word length. The resulting metric space is locally a discrete topological space, but its large-scale geometry exhibits meaningful structure. For instance, the volume of a ball of radius $r$ (i.e., the number of group elements expressible using at most $r$ generators) grows in a way that reflects intrinsic properties of the group—such as polynomial growth in nilpotent groups or exponential growth in free groups.^[1]

A finitely generated group has polynomial growth if and only if it is virtually nilpotent, and that the group's large-scale geometry can often be understood via its asymptotic cone—a kind of limiting metric space that captures the group's behavior "at infinity".

Many other aspects of infinite groups can be expressed in terms of the word metric, such as amenability: a group is not amenable, if and only if there exist constants $d$ and $C$ such that, for every finite subset $S$ , $|S|\leq C|\partial _{d}S|$ , where $\partial _{d}S$ is the set of points within a distance $d$ of any element of $S$ .

Examples

Torsion

Asymptotics

See also