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. 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.
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.
Examples for icc groups are free groups on at least two generators, or, more generally, nontrivial free products.
Famous quotes containing the words infinite, class and/or property:
“...if you are to gain any great amount of good from the world, you must attain a passive condition of mind. ...it is never to be forgotten that it is the rest of the world and not you that holds the great share of the world’s wealth, and that you must allow yourself to be acted upon by the world, if you would become a sharer in the gain of all the ages to your own infinite advantage.”
—Anna C. Brackett (1836–1911)
“During the long ages of class rule, which are just beginning to cease, only one form of sovereignty has been assigned to all men—that, namely, over all women. Upon these feeble and inferior companions all men were permitted to avenge the indignities they suffered from so many men to whom they were forced to submit.”
—Mary Putnam Jacobi (1842–1906)
“The diversity in the faculties of men, from which the rights of property originate, is not less an insuperable obstacle to a uniformity of interests. The protection of these faculties is the first object of government.”
—James Madison (1751–1836)