Carleson has proved that the Fourier series of functions belonging to the class L2 converge almost everywhere.

Improving his method, Hunt proved that the Fourier series of functions belonging to the class Lp (p > 1) converge almost everywhere. On the other hand, Kolmogoroff proved that there is an integrable function whose Fourier series diverges almost everywhere. We shall generalise Kolmogoroff's Theorem as follows: There is a function belonging to the class L(logL)p (p > 0) whose Fourier series diverges almost everywhere. The following problem is still open: whether “almost everywhere” in the last theorem can be replaced by “everywhere” or not. This problem is affirmatively answered for the class L by Kolmogoroff and for the class L(log logL)p (0 < p < 1) by Tandori.

The group G is called trifactorisable if G has three subgroups, A, B, and C such that G = AB = BC = CA. Obviously the structure of the group G will be restricted by the structure of these subgroups. In this paper it will be shown that a finite group G is π-separable if and only if it satisfies Dπ and has a trifactorisation with two factors π closed and the third, C say, π-separable. In this case we show that the π- and π′-lengths of G can be at most one more than those of C, and so it is this factor which “controls” the structure of G. Similar results are proved for π-solubility and solubility.

It is shown that rings in which the projective cover and injective hull of cyclic modules are isomorphic are equivalent to uniserial rings. Further, it is shown that rings for which the top and bottom of finitely generated modules are isomorphic also are equivalent to uniserial rings.

If the finite soluble group G admits the dihedral group of order eight as a fixed-point-free group of automorphisms then the nilpotent length of G is at most three.

There are errors in my paper [1], Lemma 1 and from the end of Lemma 4 to the end of the paper.

LEMMA 1. If L is any finite lattice and L has a point which is both join and meet reducible then L is not weakly transferable.

We replace Lemma 5 by the following sequence.

Professor William F. Reynolds has discovered a mistake in the main theorem of [1]. The problem occurs with the first display on page 339 in that is only congruent modulo radZ(G) to . The theorem still holds if θ(radZ(G)) ⊆ radZ(H), but this is not the case in general.