Let ω =(ω1) Where ω1 is a set of non-empty sets (called operations) and ω0 is a set of elements (called constants) none of which is a function whose domain belongs to ω1. An ω-ALGEBRA is a set C and a function e (the effect) from the disjoint union of ω0 and ω∈ω1Cω to C, where Cω is the set of all functions from ω to C. Let P be a set of groups Pω of permutations on ω, one group fro each ω∈ω1. A Ρ-ω-ALGEBRA is an ω-algebra such that (ρf)e = (f)e, for all ω ∈ ω1, ρ ∈ Ρω and f ∈ Cω.

Let ∧ be a K-algebra over a commutative ring K. Harada [5] has introduced the notion of weak dimension of algebras ∧ (denoted by w. dim ∧) analogous to the dimension of algebras in Cartan and Eilenberg [3].

In the present note, I investigate the question whether automorphic functions of the Fuchsian type can satisfy algebraic differential equations of the second order. Under rather general restrictions on the corresponding group, it is proved that there is no such differential equation.

Let …, αn, (n ≧ 2) be (fixed) multiplicatively independent non zero algebraic numbers and set M(H) = min |β1log α1+…+βn log αn| the minimum taken over all algebraic numbers bgr;1,…βn not all equal to zero, of degrees not exceeding a fixed natural number d0, and heights not exceeding an arbitrary natural number H. Then an important result [1] of Baker states that for every fixed ε > 0 and an explicit constant . It may be remarked that Baker deduces his general result from the special case where βn is fixed to be —1. The following straight forward generalization might be of some interest since it shows that the exponent n+1+ε need not be the best, and that the best exponent obtainable by his method has some chance of being 1 + ε (see the corollary to the Theorem).

In [2] we proved that if X admits a complete uniform structure, the intersection of the free maximal ideals in C(x) is precisely Ck(X), the ring of functions with compact support. In the present paper we are able to sharpen this result somewhat and give necessary and sufficient conditions on a space X so that this conclusion holds. Both our previous result and that of Kohls for p-spaces follow as special cases of our theorem.

It is a well-known result of W. Magnus [3] that there is a faithful matrix representation for metabelian groups i.e the groups satisfying the law ‘x, y; u, v’. The work in this paper arose in an attempt by the author to find a faithful matrix representation for centre-by-metabelian groups i.e. the groups satisfying the law ‘ x, y; u, v].

In a recent note, M. Kuczama [5] has obtained a general result concerning real solutions φ(x) on the interval 0 ≦ x < a ≦∞ of the Schröder functional equation providing the known real function satisfies the following (quite weak) conditions: f(x) is continuous and strictly increasing in ([0 a);(0) = 0 and 0 <f(x) <x for x ∈ (0, a); limx→0+ {f(x)/x} = s; and f(x)/x is monotonic in (0, a).

A subgroup A of a group G is said to be complemented in G if G contains a subgroup C such that . Every subgroup C with its property is called a complement of A in G. Various results have been obtained about groups in which each member of a given set of subgroups is complemented. Some of these results state, roughly speaking, that the existence of complements of all members of a given set of subgroups implies that all members of larger set are also complements. In this paper we derive another theorem of this kind.

The question whether any projective plane of order ten exists or not, is an unsolved problem that has attracted some interest (see, for instance, [2]). A method, by which a plane might have been discovered, was suggested to me by a theorem in [1]: ‘If order of a plane is greater than 10, a six-arc is not complete’. Elementary arguments do not, it appears, exclude the possibility of a complete six-arc in a plane of order ten: but they do show that such a figure would be of an extreme type, and that the whole plane would fit round it in a particular way. The limitation, in fact, is so severe that it becomes feasible to consider, for a good many of the incidences in the plane, all the alternative arrangements that seem possible. With the help of the Elliott 4130 computer of the University of Leicester, I have carried out an exhaustive search, and discovered that it is impossible to build up a projective plane by this method. So I can assert:

Magnus [4] proved the following theorem. Suppose that F is free group and that X is a basis of F. Let R be a normal subgroup of F and write G = F/R. Then there is a monomorphism of F/R′ in which ; here the tx are independent parameters permutable with all elements of G. Later investigations [1, 3] have shown what elements can appear in the south-west corner of these 2 × 2 matrices. In this form the theorem subsequently reappeared in proofs of the cup-product reduction theorem of Eilenberg and MacLane (cf. [7, 8]). In this note a direct group-theoretical proof of the theorems will be given.

Let X be a finite þ-group with the anti-Hughes property . (Notation is set forth in Section 2.) Any element of a finite þ-group G not in Hp (G) has order þ, hence either G/Hp(G) is non-abelian or Hp(G) ≧ (G). Therefore X has the property ; all its elements of maximal order lie in Φ (X).

Let X1, X2,…be independent and identically distributed non-lattice random variables with zero, varianceσ2<∞, and partial sums Sn = X1+X2+…+X.

Let M(t) denote the mean population size at time t (conditional on a single ancestor of age zero at time zero) of a branching process in which the distribution of the lifetime T of an individual is given by Pr {T≦t} =G(t), and in which each individual gives rise (at death) to an expected number A of offspring (1λ A λ ∞). expected number A of offspring (1 < A ∞). Then it is well-known (Harris [1], p. 143) that, provided G(O+)-G(O-) 0 and G is not a lattice distribution, M(t) is given asymptotically by where c is the unique positive value of p satisfying the equation .

Within five minutes of the start of my first meeting with Bernhard Neumann as his research student, late in 1954, he suggested the following problem to me. Let G be a group in which the cardinals of the classes of conjugate elements are boundedly finite with maximum η, say. Then the commutator subgroup G′ is finite [6], Is the order |G′| of G′ bounded in terms of n? I distinctly recall these words of Neumann: “That should provide us with a start, I think”. He was right: more than just a start, the problem has been a continuing stimulus to a study of questions in fields as far apart as permutation group theory ([7], [11], [12], [14], and some unpublished work of Peter M. Neumann) and multiplicator theory ([13], [2]), as well as attracting interest in its own right.

Given integers 0 < λ < κ < ν, does there exist a nontrivial graph G with the following properties: G is of order ν (i.e. has ν vertices), is regular of degree κ (i.e. every vertex is adjacent to exactly κ other vertices), and every pair of vertices is adjacent to exactly λ others? Two vertices are said to be adjacent if they are connected by an edge. We call a graph with the above properties a symmetric (ν, κ, λ) graph and refer to the last of the properties as the A-condition. The complete graph of order v is a trivial example of a symmetric (ν, ν— 1, ν — 2) graph, but we are of course only interested in non-trivial constructions.

A ring R is called an H-ring if for every x ∈ R there exists an interger n=n(x)> 1 such that xn-x ∈C, where C is center of R. I. N. Herstein proved that H-rings must be commutative [See 3 pp. 220–221]. We now introduce the following definition.

A group is called an ѕU-group if and only if it is locally finite and all its Sylow subgroups are abelian. Kovács [1] has shown that for any integer e the class ѕUe of all ѕ U-groups of exponents dividing е is a variety. Little is known about the laws of these varieties; in particular it is unknown whether they have finite bases. Whenever ѕUe is soluble it is an easy matter to establish explicitly a finite basis for its laws namely the exponent law, the appropriate solubility length law and all laws of the type [xm, ym]m where e = pαm, p is a prime and p does not divide m. (The significance of thelast type of law is made clear by Proposition 2 below and the obvious fact that any group that satisfies a law of this type for given prime p has abelirn Sylow p-subgroups.) For e less than thirty ѕUe is clearly soluble whikt PSL(2, 5), the non-abelian simple group of order 60, is contained in ѕU30 so that the case e = 30 is, in a sense, the first non-trivial case to be considered.

The theory of formations of soluble groups, developed by Gaschütz [4], Carter and Hawkes[1], provides fairly general methods for investigating canonical full conjugate sets of subgroups in finite, soluble groups. Those methods, however, cannot be applied to the class of all finite groups, since strong use was made of the Theorem of Galois on primitive soluble groups. Nevertheless, there is a possiblity to extend the results of the above mentioned papers to the case of Π-soluble groups as defined by Čunihin [2]. A finite group G is called Π-soluble, if, for a given set it of primes, the indices of a composition series of G are either primes belonging to It or they are not divisible by any prime of Π In this paper, we shall frequently use the following result of Čunihin [2]: Ift is a non-empty set of primes, Π′ its complement in the set of all primes, and G is a Π-soluble group, then there always exist Hall Π-subgroups and Hall ′-subgroups, constituting single conjugate sets of subgroups of G respectively, each It-subgroup of G contained in a Hall Π-subgroup of G where each ′-subgroup of G is contained in a Hall Π′-subgroup of G. All groups considered in this paper are assumed to be finite and Π-soluble. A Hall Π-subgroup of a group G will be denoted by G.

Let G be a group on two generators a and b subject to the single defining relation a = [a, ab]: . As usual [x, y] = x−1y−1xy and xy = y−1xy if x and y are elements of a group. The object of this note is to show that every finite quotient of G is cyclic. This implies that every normal subgroup of G contains the derived group G′. But by Magnus' theory of groups with a single defining relation G′ ≠ 1 ([1], §4.4). So G is not residually finite. This underlines the fact that groups with a single defining relation need not be residually finite (cf. [2]).

The contravariant functor F from the category of Riemann surfaces and analytic mappings to the category of complex algebras and homomorphisms which takes each surface Ω to the algebra of analytic functions on Ω does not have an adjoint on the right; but it nearly does. To each algebra A there is associated a surface Σ1 (A) and a homomorphism A from A into FΣ1 (A), indeed onto an algebra of functions not all of which are constant on any component of Σ1 (A), such that every such non-trivial representation A A → F(Ω) is induced by a unique analytic mapping Ω → Σ1(A)