We say that a finite group G has property N if the normalizer of every subgroup of G is normal in G. Such groups are nilpotent since every Sylow subgroup is normal (the normalizer of a Sylow subgroup is its own normalizer). Thus it is sufficient to study p-groups which have property N. Note that property N is inherited by subgroups and factor groups.

Let A be an integral domain and K its quotient field. A is called a Krull domain if there is a set {Vα} of rank one discrete valuation rings such that A = ∩αVα and such that each non-zero element of A is a non-unit in only finitely many of the Vα. The structure of these rings was first investigated by Krull, who called them endliche discrete Hauptordungen (4 or 5, p. 104).

In (1) the author suggested the following research problem. Does there exist a non-periodic entire function ƒ such that ƒƒ is periodic? My aim in this note is to give a partial answer to this question and, more generally, to give a partial solution to the following problem: if ƒ and g are entire functions and ƒ(g) is periodic, what can one say about g? These results also extend a previous result of mine; for details, see (2, Theorem 4). We begin with some simple lemmas.

A abelian group difference set (abbreviated AGDS) (G, D) is a -subset D = {di}1k taken from an abelian group G of order v such that each element different from the identity e in G appears exactly λ times in the set of differences {didj-1}, where . Combinatorially, AGDS is equivalent to a design having an abelian collineation group which is transitive and regular on the elements and on the blocks of the design (1).

It was shown in (1) that a complete set of bounded observables is metrically complete. However, an extra axiom was needed to prove this result (1, footnote, p. 436). In this note we prove the above-mentioned result without the extra axiom. We also show that there is an abundance of pure states if M is closed in the weak topology and give a necessary and sufficient condition for the latter to be the case.

It has been shown by Walsh (3) and Szegö (2) that if a set of polynomials is orthogonal on both of two distinct curves, then one curve is a level curve of the other. Szegö (2) has determined all sets of polynomials which are orthogonal simultaneously on an entire family of level curves. There are five essentially different sets, two of which are orthogonal on concentric circles, and three of which are orthogonal on confocal ellipses. Merriman (1) has shown that the orthogonality of a set of polynomials on both of two concentric circles is sufficient to guarantee their orthogonality on the entire family of circles.

In (1), Berstein and Ganea denned the nilpotency class of a based topological space. For a based topological space X we write nil X for the nilpotency class of the group ΩX in the category of based topological spaces and based homotopy classes. Hilton, in (3), defined the nilpotency class, nil class K of a based semi-simplicial (s.-s.) complex; actually, the restriction of connectedness can be removed. Hence, by using the total singular complex functor S, an invariant (nil class SX) can be defined for a based topological space X.

A finite group G is said to be a fixed-point-free-group (an FPF-group) if there exists an automorphism a which fixes only the identity element of G. The principal open question in connection with these groups is whether non-solvable FPF-groups exist. One of the results of the present paper is that if a Sylow p-group of the FPF-group G is the direct product of any number of mutually non-isomorphic cyclic groups, then G has a normal p-complement. As a consequence of this, the conjecture that all FPF-groups are solvable would be true if it were true that every finite simple group has a non-trivial SylowT subgroup of the kind just described. Here it should be noted that all the known simple groups satisfy this property.

In (1) Armentrout raised the question “Is there a monotone decomposition of E3 into arcs?” The analogous question for E2 was answered negatively by Roberts in (8). Our aim in this paper is to give a partial answer to Armentrout's question by proving the following theorem.

Let A and H be finite groups with A acting on H, i.e., there is a given, fixed homomorphism A → Aut(H). In this situation, A acts on the set of conjugacy classes of H and also on the set of irreducible characters of H. If A is cyclic, it follows from a lemma of Brauer (see, for instance, 1, 12.1) that the number of fixed points in these two actions are equal and therefore one can conclude that for any A, the number of orbits in the two actions are equal.

It is a well-known result of Sanov (5) that groups of exponent pk (p prime) satisfy the th Engel congruence (definition below). Recently, an alternative proof of this has been given by Glauberman, Krause, and Struik (3). Bruck (2) has conjectured that such groups satisfy the th Engel congruence. In this note we go some way towards proving this.

The series is said to be summable (L) to s if the sequence {sn}, where sn = a0 + a1 + … + an, is L-convergent to s, i.e., if

If the sequence {sn} is l-convergent to s, i.e., if the sequence {tn}, where

1

Unless stated otherwise, all our matrices (denoted by capital letters) are square matrices of size n × n and composed of real numbers. A' denotes the transpose of A. The characteristic or eigenvectors of matrices are written as column vectors having n coordinates. If ζ is a vector, ζ’ denotes its transpose.

In this paper we describe an algorithm for finding the number of non-isomorphic -coloured graphs with n nodes and e edges. We use Pόlya's fundamental enumeration theorem (in a form similar to that given by de Bruijn (see 1)) which reduces the problem to finding the cycle index for a certain permutation group. Harary (3) followed this same program for bi-coloured graphs, but failed to find the cycle index of the relevant group for general -coloured graphs.

In this note the Hilbert spaces under consideration are complex, and the operators referred to are bounded, linear operators. If is a Hilbert space, then the algebra of all operators on is denoted by .

It is known (1) that if is any Hilbert space, then the class of commutators on , i.e., the class of all operators that can be written in the form PQ — QP for some , can be exactly described. A similar problem is that of characterizing all operators on that can be written in the form for some .

A subset {s α| α ϵ △} of a lattice-ordered group (l-group) is disjoint if S α Λ S β = 0 for all α≠ β in △. An l-group G has the lifting property if for each l-ideal S of G and each countable disjoint subset X1 , X2, … of G/S one can choose elements 0 ≦ xi ϵ Xi so that x1, x2, … is a disjoint subset of G. In (2) Topping showed by an example, that uncountable sets of disjoint elements cannot necessarily be lifted and asserted (Theorem 8) that each vector lattice has the lifting property. His proof is valid for finite disjoint subsets of G/S, but we show by an example that this is, in general, all that one can establish.

Let G be a bounded open set of Rn with a smooth boundary ∂G. We consider the following elliptic boundary-value problem:

where A and Bj are, respectively singular integro-differential operators on G and on ∂G, of orders 2m and rj with rj < 2m; Ck are boundary differential operators, and Ljk are linear operators, bounded in a sense to be specified.

Notation and definitions.

Definition 1. Let (X, ρ) be a metric space and ϕ: X → X a continuous self-mapping of X. We shall call ϕ and α-contraction on (X, ρ) if and only if α ϵ [0,1) and . We shall call ϕ an α-homothety on (X, ρ) if and only if α > 0 and .

Let X be a topological space and ϕ: X ⟶ X a continuous self-mapping of X. We say that ϕ is linearized in L by Φ if there exists a topological embedding μ: X ⟶ L of the space X into the linear topological vector space L such that for all x ϵ X, μ (ϕ (x)) = Φ (μ (x)), where ϕ is a continuous linear operator on L.

In § I, it is shown that M(G)*, the space of bounded linear functionals on M(G), can be represented as a semigroup of bounded operators on M(G).

Let △ denote the non-zero multiplicative linear functionals on M(G) and let P be the norm closed linear span of △ in M(G)*. In § II, it is shown that P, with the Arens multiplication, is a commutative B*-algebra with identity. Thus P = C(B), where B is a compact, Hausdorff space.