The structure of the group ring of a $p$-group over a modular field

This paper deals with the group ring of a group of prime power order over the prime field GF(p), where p is the prime dividing the order of the group. It is well known that in the case of the group ring of a group over a field whose characteristic divides the order of the group, the ordinary theory of group characters is no longer valid: recently, Brauer and Nesbitt(') have investigated the properties of the modular representations in this case, but this general theory yields only little in the special case that we consider here. We investigate the group ring from the point of view of the structure of its radical, and in particular, determine a basis for, and the ranks of, the various powers of the radical in terms of the elements and order of a new series of characteristic subgroups. These subgroups are defined by a certain minimal property which combines the commutator and the pth power structure of the group, and should prove useful in general investigations on the structure of p-groups. 1. It is well known that the group ring of a group of order g is semisimple, provided the characteristic of the underlying field is zero, or a prime which does not divide g(2). If, however, the underlying field has characteristic p, and p divides g, then it is readily seen that the group ring has a radical which is not zero. Let the elements of the group be G1= 1, G2, * * *, G,. Consider the element a=Gi+G2+ +G, in the group ring. We have -G,=, and hence, if A =>aiGi is any element in the ring, a . A = A o a= ( Eai) o ; that is, scalar multiples of a form an ideal (a). However, (a). (a) = 0, since o*.f* -* (E1) =0, ((:1) =g-O modulo p), and hence the group ring contains a nilpotent ideal different from zero. We have proved(3), therefore,