The purpose of this paper is to prove that o(l/x) is the best possible Tauberian condition for the collective continuous Hausdorff method of summation. The analogue of this result for the collective (discrete) Hausdorff method is known [1, pp. 229, ff.; 7, p. 318; 8, p. 254]. Our method involves generalizing a well-known Abelian theorem of Agnew [2] to locally compact spaces and then applying the analogue for integrals of a result Lorentz obtained for series [6, Theorem 1].

Let T and X denote locally compact, non compact, σ-compact Hausdorff spaces. Let T′ = T ∪ (∞) and X′ = X ∪ (∞) denote the onepoint compactifications of T and X, respectively. Let B(T) denote the set of locally bounded, complex valued Borel functions on T and let B∞(T) denote the bounded functions in B(T).

Throughout this paper G will denote a finite group containing a nilpotent maximal subgroup S and P will denote the Sylow 2-subgroup of S. The largest subgroup of S normal in G will be designated by core (S) and the largest solvable normal subgroup of G by rad(G). All other notation is standard.

Thompson [6] has shown that if P = 1 then G is solvable. Janko [3] then observed that G is solvable if P is abelian, a condition subsequently weakened by him [4] to the assumption that the class of P is ≦ 2 . Our purpose is to demonstrate the sufficiency of a still weaker assumption about P.

A (non-archimedean) valuation v on a field K is said to be henselian if it has a unique prolongation to a valuation on Ka, the algebraic closure of K. A henselization (Kh, vh) of a valuated field (K, v) is a smallest separable extension of K containing a henselian prolongation vh of v. (Kh, vh) is unique up to K-isomorphism, and (Kh, vh) = (K, v) if and only if v is henselian. In this paper we confine ourselves to valuations of finite rank.

If v is a non-henselian rank one valuation on K, and if [Ks:K] = ∞, Ks being the separable closure of K, then it is known that v has infinitely many prolongations to Ka [1, (27.11)]. We shall see that this is no longer true if the rank of v is greater than one.

In 1955 Taylor [6] constructed an AP-integral sufficiently strong to integrate Abel summable series with coefficients o(n). He showed that the AP-integral includes the special Denjoy integral and further that, when applied to trigonometric series, the AP-integral is more powerful than the SCP-integral of Burkill [1] and the P2-integral of James [3]. The present paper shows that the AP-integral includes the SCP-integral, and, under natural assumptions, the P2-integral.

After completing this manuscript I was advised by Skvorcov that he had shown [5] under more general conditions that the P2-integral is included in the AP-integral. The proof in the present paper seems to have some value in its own right and is considerably shorter.

Since the definition of the AP-integral is essentially for a function defined in (0, 2π] and elsewhere by 2π-periodicity, we shall consider SCP-integrable and P2-integrable functions defined similarly.

Let H be a subgroup of a group G (all groups considered throughout this article are finite); then H will be called primitive if the subgroup

is distinct from H. Such subgroups, which are also called meet-irreducible, arise naturally in connection with minimal permutation representations of groups and in other contexts; for example, every subgroup of a group G can be written as an intersection of primitive subgroups of G, and the set of all primitive subgroups of G is characterized by its minimality with respect to this property. While maximal subgroups are always primitive, most groups contain non-maximal subgroups which are primitive (see remark at end of article). Note that a subgroup H of an abelian group G is primitive if, and only if, G/H is cyclic of prime-power order.