Let B be a topological algebra with Fréchet space topology, A an algebra with locally convex topology and an algebra of formal power series over A in n commuting indeterminates which carries a Fréchet space topology. In a previous paper the author showed, for the case n = 1, that a homomorphism of B into whose range contains polynomials is necessarily continuous provided the coordinate projections of into A satisfy a certain equicontinuity condition. This result is here extended to the case of general n, and also to weaker topological assumptions.

A number of questions of Philip Hall concerning complemented normal subgroups of finite relatively free groups are considered.

The class of ordered topological spaces which are projective limits of finite partially ordered sets (equipped with the restriction of the product of the discrete topologies) is shown to coincide with the class of compact totally order-disconnected ordered topological spaces. Hence this is another category of spaces equivalent to the category of distributive lattices with zero and unit.

This note establishes the impossibility of certain inequalities of the form holding for all trigonometric polynomials f on an infinite compact abelian group G. From this is deduced the impossibility of corresponding inclusion relations of the type or where FS denotes the Fourier image of the set S of integrable functions on G.

The topology of a topological group in which the intersection of open sets is open is uniquely determined by a normal subgroup, and the group is uniquely an extension of an indiscrete group by a discrete group. This was proved by M.C. Thornton under the additional hypothesis that the group is a torsion group. The proofs here given make the more general facts almost trivial.

It is shown that there exists a logarithmic upper bound on the Fitting length h(G) of a finite soluble group G in terms of the number ν(G) of the conjugacy classes of its maximal nilpotent subgroups. For ν(G) = 3, the best possible bound on h(G) is shown to be 4.

In J. Math. Mech. 15 (1966), 877–898, Bonic and Frampton have laid the foundation for a general theory of smoothness of Banach spaces. In this paper, we shall study one aspect of the smoothness of topological vector spaces, namely, the relationship between smoothness and inductive and protective limits of topological vector spaces. As a consequence, we obtain smoothness results for nuclear spaces and some Montei spaces.

We consider a semigroup S acting as affine continuous maps on a compact convex set X. F denotes the corresponding set of fixed points. Let exX and exF denote the corresponding sets of extreme points. If X is a simplex, conditions are given which ensure that when x ε F, the maximal measure representing x invariant under S. We also prove exF = F ∩ exX under conditions involving extreme amenability of S. Topological properties of exF are also studied.

We give another formulation of Theorem 1 of Ribenboim's paper, Bull. Austral. Math. Soc. 2 (1970), 81–88. Our condition on a module to be totally ordered such that this order extends a given strict order seems to be more useful for applications than condition (2) of Ribenboim. This is shown for example in Corollaries 3 and It of our paper. Corollary 4 is due to Ribenboim. Corollary 3 generalizes the corresponding well known result on abelian groups.

A characterization of the maximal normal p-nilpotent subgroup of a finite p-solvable group is obtained for primes p ≠ 2 or 3.

A theorem proved by de Leeuw for Γ = Rn and later generalized by Lohoué and Saeki states that if Γ is an LCA group, Γ0 a closed subgroup thereof, π the canonical mapping from Γ onto Γ/Γ0 and Φ a Fourier multiplier of type (p, p) on Γ/Γ0, then Φ ० π is a Fourier multiplier of type (p, p) on Γ. We show here that if 1 ≤ p < q ≤ ∞, Γ and Φ is a Fourier multiplier of type (p, q) on Γ/Γ, then Φ ० π is a Fourier multiplier of type (p, q) on Γ and if Γ0 is a non-compact subgroup of Γ and Φ ० π is a Fourier multiplier of type (p, q) on Γ for some p and q satisfying 1 ≤ p ≤ q ≤ ∞, then Φ is zero. We prove also that if Φ is a Fourier multiplier of type (p, q) and Γ/Γ0, whee 1 ≤ q < p ≤ ∞ and Γ is discrete, then Φ ० π is a Fourier multiplier of type (p, q) on Γ.

Let G be a group with a normal nilpotent subgroup N such that G/N is periodic and nilpotent. If G(p)/N is the Sylow p-subgroup of G/N and Q(p) is the maximal p-radicable subgroup of N, it is shown that G has a bound on the subnormal indices of its subnormal subgroups if and only if there is a positive integer c such that G(p)/Q(p) is nilpotent of class at most c, for all primes p. It is also shown that if G is a periodic metanilpotent group and Q is its maximal radicable abelian normal subgroup then G has a bound on its subnormal indices if and only if there is a positive integer c such that for all primes p the Sylow p-subgroups of G/Q are nilpotent of class at most c.

Led on by a paper of Brown and Dancer with the same title, this paper gives two further theorems on asymptotic expansions of composite functions.

We construct a group G with the properties stated in the title.

The statements of Lemma 3 and Lemma 4 in [1] are incorrect. They should read as follows.

LEMMA 3.

LEMMA 4.