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 I could only live at the pitch that is near madness
When everything is as it was in my childhood
Violent, vivid, and of infinite possibility:
That the sun and the moon broke over my head.”
—Richard Eberhart (b. 1904)
“The ideas of the ruling class are in every epoch the ruling ideas, i.e. the class which is the ruling material force of society, is at the same time its ruling intellectual force.”
—Karl Marx (1818–1883)
“Oh, had I received the education I desired, had I been bred to the profession of the law, I might have been a useful member of society, and instead of myself and my property being taken care of, I might have been a protector of the helpless, a pleader for the poor and unfortunate.”
—Sarah M. Grimke (1792–1873)