On the normal complement of G in the group of units V(F_pⁿG) in group algebras of a finite p-groups
by
László Erdei
University of Debrecen, Debrecen

Let F_pⁿG be the group algebra of a finite p-group G over a field of pⁿ elements, and V(F_pⁿG) the subgroup of units of augmentation 1. Then V(F_pⁿG) is a finite p-group of order p^n(|G|-1). One of the hardest and most important problems, which was proposed by Dennis is the following: for which modular group algebras F_pⁿG exists a normal complement subgroup N for G in V(F_pⁿG).

In 1980, Ivory gave the first and so far the only counterexamples: if p=2 and G is a 2-group of maximal class with order greater than 8, then G cannot have a normal complement in V(F₂G). For a nonabelian group G of order 8 we show that there does not exist a normal complement in V(F_2ⁿG) for n > 1 and we describe the unit group V(F_2ⁿG).

Let V_* be the subgroup consisting of the elements of V that are inverted by *. These elements are called the unitary units of F_p G. Naturally, G <= V_* <= V. If G has a normal complement in V then obviously it also has a normal complement in V_*. However, V. Bovdi and T. Rozgonyi proved that if G is any generalized quaternion group then it does have a normal complement in V_*. This raised the possibility that perhaps every G has a normal complement in V_*. Our joint results with A. Bovdi show that this is not the case: we proved that if G is a dihedral 2-group or a semidihedral 2-group and if the order of G is greater than 16, then G does not have a normal complement in V_*.

Date received: July 31, 2000