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.

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.

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].

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.

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).

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 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.

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]).

In [2], John Cossey and Sheila Oates Macdonald give a basis for the set of laws of PSL(2, 5) — the simple group of order 60 — and with one extreme exception the laws of their basis involve at most two variables. They raise the problem of finding a basis in which all of the laws involve only a small number of variables, and remark that they have shown that five variables will suffice. Here we give a basis consisting entirely of two variable laws.

The purpose of this paper is to improve a result in [2] on the automorphisms of the semigroup D = D(E) of all (Fréchet)-differentiable mappings of a real Banach space E into itself.