In group theory, an area of mathematics, an infinite group is a group whose underlying set contains infinitely many elementss. 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 or free groups.

Structure and classification

Many infinite groups are given in terms of a set of generators and relations. For example, a free group is a group on a set of generators with no relations, whereas a braid group.

or any Lie group; finitely presented such as any free group (on a finite set of generators) or braid groups. Groups may also be infinitely presented, etc.

An infinite group may be residually finite, are residually finite; whereas others like the Tarski monster groups, are not.

Many infinite groups are linear groups and every classical group (via its adjoint representation), and every finitely-generated torsion-free nilpotent group, by Malcev's theorem, but not groups like the metaplectic group.

Some infinite groups are simple, such as the Thompson groups.

Asymptotics

inherits a natural metric structure via the word metric, and distances extend by minimal word length. The resulting metric space is locally a discrete topological space 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.

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.

See also

• Borel equivalence relation

• Descriptive set theory

• Finitely generated group

• Infinite-dimensional Lie group

• Locally compact group

• Polish group

References