A Conjecture of Brian Hartley and developments arising

Abstract

Around 1980 Brian Hartley conjectured that if the unit group of a torsion group algebra FG satisfies a group identity, then FG satisfies a polynomial identity. In this short survey we shall review some results dealing with the solution of this conjecture and the extensive activity that ensued. Finally, we shall discuss special polynomial identities satisfied by FG (or by some of its subsets) and the corresponding group identities satisfied by its unit group (or by some of its subsets).