In mathematics, a group is said to have the infinite conjugacy class property, or to be an ICC group, if the conjugacy class of every group element but the identity is infinite.Template:R
The von Neumann group algebra of a group is a factor 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.<ref>Template:Citation. See in particular p. 450: "LΓ is a II1 factor iff Γ is ICC".</ref>
Examples of ICC groups are the group of permutations of an infinite set that leave all but a finite subset of elements fixed,Template:R and free groups on two generators.Template:R
In abelian groups, every conjugacy class consists of only one element, so ICC groups are, in a way, as far from being abelian as possible.