# Infinite conjugacy class property

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.[1]

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.[2]

Examples for icc groups are the group of permutations of finitely many elements of an infinite set,[3] and free groups on two generators.[3]

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.

## References

1. ^ Palmer, Theodore W. (2001), Banach Algebras and the General Theory of *-Algebras, Volume 2, Encyclopedia of mathematics and its applications, 79, Cambridge University Press, p. 907, ISBN 9780521366380.
2. ^ Popa, Sorin (2007), "Deformation and rigidity for group actions and von Neumann algebras", International Congress of Mathematicians. Vol. I (PDF), Eur. Math. Soc., Zürich, pp. 445–477, doi:10.4171/022-1/18, MR 2334200. See in particular p. 450: "LΓ is a II1 factor iff Γ is ICC".
3. ^ a b Palmer (2001), p. 908.