In this short note, we show that the class of abelian groups determined by the subgroup lattice of their direct n-powers is exactly the class of the abelian groups which share the n-root property. As applications we answer in the negative a (semi)conjecture of Palfy and solve a more general problem. Recently, for an arbitrary group G, the subgroup lattice of the square G×G has received some att...

A countable group is C∗-simple if its reeduced C∗-algebra is a simple algebra. Since Powers recognised in 1975 that non-abelian free groups are C∗-simple, large classes of groups which appear naturally in geometry have been identified, including non-elementary Gromov hyperbolic groups and lattices in semisimple groups. In this exposition, C∗-simplicity for countable groups is shown to be an ext...