On the unit conjecture for supersoluble group algebras

Abstract

We introduce structure theorems for the study of the unit conjecture for group algebras of torsion-free supersoluble groups. Motivated by work of P.M. Cohn we introduce the class of (X,Y,N)-group algebras KG, and following D.S. Passman we define an induced length function L:KG→N∪{−∞} using the fact that G has the infinite dihedral group as a homomorphic image. We develop splitting theorems for (X,Y,N)-group algebras, and as an application show that if σ∈KG is a unit, then L(σ)=L(σ−1). We extend our analysis of splittings to obtain a canonical reduced split-form for all units in (X,Y,N)-group algebras. This leads to the study of group algebras of virtually abelian groups and their representations as subalgebras of suitable matrix rings, where we develop a determinant condition for units in such group algebras. We apply our results to the fours groupΓ=〈x,y|xy2x−1=y−2,yx2y−1=x−2〉 and show that over any field K, the group algebra KΓ has no non-trivial unit of small L-length. Using this, and the fact that L is equivariant under all KΓ-automorphisms obtained K-linearly from Γ-automorphisms, we prove that no subset of the Promislow set P⊂Γ is the support of a non-trivial unit in KΓ for any field K. In particular this settles a long-standing question and shows that the Promislow set is itself not the support of a unit in KΓ. We then give an introduction to the theory of consistent chains toward a preliminary analysis of units of higher L-length in KΓ. We conclude our work showing that units in torsion-free-supersoluble group algebras are bounded, in that the supports of units and their inverses are related through a property (U) and the induced length function L.