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.
It will then be, provided the group is nontrivial, of type II1, i.e. it will possess a unique, faithful, tracial state.
[2] Examples of ICC groups are the group of permutations of an infinite set that leave all but a finite subset of elements fixed,[1]: 908 and free groups on two generators.
[1]: 908 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.
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