Infinite conjugacy class property

ICC groups in mathematics studied groups with infinite conjugacy classes. The abbreviation ICC stands for the English name infinite conjugacy classes.

Definition

A group with at least two elements is called ICC - group if each of several conjugacy class is infinite, wherein the neutral element of the group was.

This means that for each element of the set is infinite.

Comments

• ICC groups are infinite and highly non- commutative. The center of an ICC group consists only of the identity element.
• The left regular representation of a discrete ICC group to be produced on a type II1 factor. Hence its importance.

Examples

• The free group with many producers is an ICC group.
• The group of finite permutations of ICC is a group, it is the group of the subgroup generated by all transpositions on the full permutation group.
• Cartesian products of finitely many ICC groups are ICC groups again.