In mathematics, a group is said to have the infinite conjugacy class property, or to be an ICC group, if the

conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other w ...

of every group element but the identity is

infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group) Infinite ( ko, 인피니트; stylized as INFINITE) is a South Ko ...

. The von Neumann group algebra of a group is a

factor Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, ...

if and only if the group has the infinite conjugacy class property. It will then be, provided the group is nontrivial, of type ''II₁'', i.e. it will possess a unique, faithful, tracial state.. See in particular p. 450: "''L''Γ is a II₁ factor iff Γ is ICC". Examples of ICC groups are the group of

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...

s of an infinite set that leave all but a finite subset of elements fixed,, p. 908. and

free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''− ...

s on two generators. In

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...

s, every conjugacy class consists of only one element, so ICC groups are, in a way, as far from being abelian as possible.

References

Infinite group theory Properties of groups {{algebra-stub