In this note we give a basis for the radical of the group algebra of a p-nilpotent group over a field of characteristic p in terms of the ordinary representation theory of the group. We use our result to calculate the exponent of the radical for such a group.

A Sasakian space [1]Mn (n = 2m + 1) is a Riemannian n-space with a positive definite metric tensor gij and a unit Killing vector field η which satisfies where the comma denotes covariant differentiation with respect to the metic tensor. In a recent paper [2] M. C. Chaki and A. N. Roy Chowdhury studied conformally recurrent spaces of second order, or briefly conformally 2-recurrent spaces, that is, non-flat Riemannian spaces Vn (n > 3) defined by where is the conformal curvature tensor: and alm is a tensor not identically zero.

All rings considered are to be associative. For definitions not included in this paper see [2]. Let R be a ring and S a subring of R.

I. Šafarevič [2] proves that for a pro-finite p-group G, the deficiency of G (in the topological sense) equals the negative of d(m(G)). This means that if G is a finite p-group, where d(m(G)) = n, then there exists a group K with deficiency −n, such that G is the maximal p factor of K.

Finite groups in which normality is transitive have been studied by Best and Taussky, [1], Gaschütz, [3], and Zacher [16]. Infinite soluble groups in which normality is transitive have been studied by Robinson in [9]. A subgroup H of a group G is subnormal in G if H can be connected to G by a chain of r subgroups, in which each is normal in its successor, where r is a non-negative integer. The least such r is called the subnormal index of H in G (or the defect of H in G). Then groups in which normality is transitive are precisely those in which every subnormal subgroup has subnormal index at most one. Thus the structure of soluble groups in which every subnormal subgroup has subnormal index at most n (such a group is said to have bounded subnormal indices) has been dealt with by Robinson in [9] for the case where n is one. However Theorem D of [12] states that a soluble group of derived length n can be embedded in a soluble group in which the subnormal indices are at most n. Therefore we must impose further conditions on the groups if we hope to obtain any worthwhile results for the above problem with n greater than one.

Let E1 and E2 be real Banach spaces and let L(E1) and L(E2) be the Banach algebras of all continuous linear mappings on E1 and E2 respectively. It is a well- known result of M. Eidelheit [1] that L(E1) that L(E2) are isomorphic as rings if and only if E1 and E2 are topologically and algebraically isomorphic. It is easy to see that the essential part of his proof is the following fact.