The set of rational points forms an infinite abelian group, which shall be called G in this article.

These groups are our first examples of infinite non-abelian groups.

His first paper was published at 19 on infinite abelian groups.

This is the Coxeter complex of the infinite dihedral group.

On the other hand, the theory of infinite abelian groups is an area of current research.

Among infinite groups, linear groups form an interesting and tractable class.

For example, in the infinite dihedral group, which has presentation:

However this group is the direct product of two infinite cyclic groups and so has solvable word problem.

All other finitely-generated infinite groups have exactly one end.

It is much more difficult to construct finitely generated infinite simple groups.