On group algebras with unit groups of derived length at most four

Abstract

In this paper, we examine the relation between a finite group G and the units U in the group algebra of G over a field K of positive characteristic. By imposing certain natural conditions on the derived subgroups of U so that it has solvable length at most four, we show that the group G must be commutative.

On the derived length of units in group algebra

“…In this paper we extend the result from torsion nilpotent groups to groups G with the unit group U = U (KG) in KG having derived length at most four. In an earlier paper [11], we discussed a special case when (U (3) , U ′ ) = {1} and G is of odd order using a combinatorial argument after considering several cases and subcases. In the present paper we resolve it completely for any G with U = U (KG) of derived length four without any condition on the order of G. The present argument is far simpler and in particular, it avoids the combinatorial argument altogether.…”