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,