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.

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 ${\displaystyle \mathbb {Z} }$ or free groups.

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 is a group on generators ${\displaystyle g_{i}}$, where ${\displaystyle i}$ is an integer in ${\displaystyle [0,n]}$, with relations ${\displaystyle g_{i}g_{i+1}g_{i}=g_{i+1}g_{i}g_{i+1}}$ and ${\displaystyle g_{i}g_{j}=g_{j}g_{i}}$ if ${\displaystyle |i-j|>1}$.

Infinite groups can be finitely generated, such as ${\displaystyle \mathbb {Z} }$ or ${\displaystyle \operatorname {SL} (n,\mathbb {Z} )}$, or infinitely generated such as ${\displaystyle (\mathbb {Q} ,+)}$ 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, meaning that every element is non-trivial in some finite quotient. Many groups, like ${\displaystyle \operatorname {SL} (n,\mathbb {Z} )}$, are residually finite; whereas others like the Tarski monster groups, are not.

Many infinite groups are linear groups, meaning that they have a faithful representation on a finite-dimensional vector space. This includes groups like ${\displaystyle \operatorname {SL} (n,\mathbb {Z} )}$ 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.

An infinite group equipped with a generating set ${\displaystyle \{\gamma _{i}\mid i\in I\}}$ inherits a natural metric structure via the word metric. This is the unique left-invariant distance function such that ${\displaystyle d(\gamma _{i},e)=d(\gamma _{i}^{-1},e)=1}$ for each generator ${\displaystyle \gamma _{i}}$, 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 ${\displaystyle r}$ (i.e., the number of group elements expressible using at most ${\displaystyle r}$ 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.