1. A group G is called characteristically simple if it has no proper non-trivial subgroups which are invariant under all automorphisms of G. It is known that if G is characteristically simple then each countable subgroup lies in a countable characteristically simple subgroup of G. A similar assertion holds for simple groups. These results were proved by Philip Hall in lectures in 1966, and further proofs appear in (4) and (6). For simple groups there is a well known and elementary result in the other direction: if every two-generator subgroup of a group G lies in a simple subgroup, then G is simple. These considerations prompt the question (first raised, I believe, by Philip Hall) whether a group G is necessarily characteristically simple if each countable subgroup lies in a characteristically simple subgroup.

As is well known, there is an algorithm for deciding if a system of linear equations with coefficients from the set Z of integers has a solution in integers. The purpose of this paper is to answer the following question: does this remain true if ‘linear’ is replaced by ‘quadratic’?

1. The equation of the scroll R of tangents of the curve Γ of intersection of two quadrics in general position in [3] was found by Cayley in 1850 ((l), p. 50). Cayley also considered special positions of the quadrics but we shall not be concerned with these. was reproduced in Salmon's treatise ((8), p. 191) and Salmon, on this same page, provides an equation in covariant form and so applicable to pairs of quadrics whether in canonical form, or in general position, or not.

1. Introduction. It is well known (see (5) and (6)) that if T is a 2-group containing an involution t such that |CT(t)| = 4, then T is dihedral, semidihedral or cyclic of order 4. Fomin(1) has studied 2-groups T containing an involution t such that |CT(t)| ≤ 8. He showed in particular that such a group has sectional 2-rank at most 4. In this paper we will prove the following:

Theorem A. Let T be a 2-group containing an involution t such that |CT(t)| ≤ 16. Then T has sectional 2-rank at most 6.

For numbers of increasing real functions f(x) with new integral inequalities. They generalize classical results. The proofs are short and simple being based on sequences.

We shall be concerned with the Laguerre polynomials, defined by Some time ago Szegö (8) showed that for (a, b, c = 0, 1, …). It is easily deduced from this (cf. (5)) that for all A λ ≥ 3.

1. Introduction. In (2) we associated with a commutative ring R with identity the lattice ℒ(R) of rings S which satisfy (i) R ⊆ S ⊆ T, where T is the total quotient ring of R, and (ii) dS ⊆ R for some d ∈ R which is not a zero-divisor. The rings of ℒ(R) can be constructed by blowing up ideals of the set of ideals A of R which satisfy (i) 0: A = 0, and (ii) there exist integers m1n ≥ 1 and α ∈ An such that αAm = Am+n (see Propositions 2·3 and 2·5 of (2)).

In this paper, a new family of factors of (2, 3, 7) is obtained which contains groups found in three other papers. It is shown that for all n, there exists an m such that there are at least n isomorphism types of Hurwitz groups of order m. Finally, presentations for all groups considered are obtained.

A k-3 cap in a three-dimensional Galois space, S3,q, is a set of k points, of which some 3, but no 4 are collinear. It is shown in (2) that for q ≥ 4 .

1. Introduction. Let X be a space and let E be a fibre space over E. A fibre-preserving map f: E → E determines, for each point x ∈ X, a map fx: Ex → Ex of the fibre over x. In a previous note (3) the situation was considered where fx is null-homotopic, for all x. In the present note we turn our attention to the situation where fx is homotopic to the identity on Ex, for all x ∈ X. If X admits a numerable categorical covering (as when X is an ANR) then such a fibre-preserving map f is a fibre homotopy equivalence, by the well-known theorem of Dold(1). Then the set Φ1(E) of fibre homotopy classes of such maps forms a normal subgroup of the group Φ*(E) of fibre homotopy classes of fibre homotopy equivalences. The purpose of this note is to prove

Theorem 1.1. Let X be a paracompact space of finite category. Let E be a fibre bundle over X of which the fibres are compact and path-connected ANR's. Then the Φ*(E)-growp Φ1(E) is Φ*(E)-nilpotent of class ≤ cat X.

Every real number α admits many rational approximations in the sense that for every natural number q,

for some integer p. Perhaps surprisingly, numbers α which have k distinct rational approximations with given denominators q1,…,qk,

for a fixed ε > 0, are quite sparse; their measure in U = [0,1) is at most C(ε)/k.

This note approaches the immersion/embedding problem for real projective n-space Pn in Rn+k by viewing the Smale invariant of the induced immersion Sn → Pn → Rn + k as an obstruction to extending to an immersion of Pn + 1 in Rn + k. First steps in this direction were taken in (8), by studying the elementary problem of extending axial maps.

Let G bea finite group, let C1, …, Cn be the cyclic subgroups of G and let 1i be the identity linear character of Ci (1 ≤ i ≤ n). It is a well-known result of Artin ((l), 39·1) that a character x of a linear representation of G over the rational numbers may be written

where a1, …, an are integers. In this note, we establish an analogue of this result for characters of projective representations over the rational numbers.

A symmetric operator on a Hilbert space, with deficiency indices (m; m) has self-adjoint extensions. These are ‘highly reducible’. The original operator may be irreducible, (see example (i), below). Can the mechanism whereby reducibility is achieved be understood? The concrete examples most readily studied are those associated with differential operators. It is easy to obtain operators, associated with a formal linear differential operator, having deficiency indices (m; m). What of reducibility? Nothing seems to be known. In the case of the first-order operator we were able, using the Volterra operator, to establish irreducibility of the associated minimal operator. To investigate symmetric operators associated with a second-order differential operator, different methods had to be developed. They apply also to the first-order operator, and we employ them to demonstrate the irreducibility of the associated minimal operator. In the second-order case the minimal operator proves reducible, and we also exhibit examples of reducibility of associated symmetric operators. It would clearly be of interest to elucidate the influence of the boundary conditions on reducibility.

For any group G and x ∈ G, and any n ∈ ℕ (the natural numbers) let

i.e. the set of all nth roots of x in G.

In (4) we showed how to stably split certain spaces CX built up from a ‘coefficient system’ and a ‘Π-space’ X. Via an approximation theorem relating particular examples to loop spaces, there resulted stable splittings of Ωn σnX for all n ≥ 1 and all (path) connected based spaces X.

In (3), corollary, p. 373) Burch gives the following inequality for the analytic spread l(I) of an ideal I of a noetherian local ring (R, m):

In this paper we shall improve this by showing that the number min depth (R/In) may be replaced by the asymptotic value of depth (R/In) for large n (which exists) (see Section (2)). By its definition (see (6), def. 3)) the analytic spread is of asymptotic nature, i.e. depends on the modules In/mIn = Un only for large n. We shall prove a stronger result, Section (4), which also shows the asymptotic nature of l(I). This result might be interesting for itself, particularly as it is not of local nature. Once Section (4) is proved and once we know that depth (R/In) is asymptotically constant (which turns out to be an easy consequence of (1), (1)), our improved inequality is easily established: Indeed, replacing R by R/xR where x is regular with respect to almost all modules (R/In), we perform a change which affects only finitely many of the modules Un (see Section (8)).

An element B ∈ Bn, the braid group on n strings, which can be written as

in terms of the standard generators of Bn is called a split braid. It is easy to see that the resulting closed braid is the connected sum of the closed braids Ĉ and on k + 1 and n − k strings respectively (figure 1).

This article is concerned with the bifurcation of critical points of a smooth real-valued function on a Banach space. There are two somewhat different aspects considered. The first, referred to as reduction, is a refinement of the Liapunov-Schmidt procedure by incorporating techniques from the theory of determinate germs and their universal unfoldings. A definite reduction procedure is given, whereby a germ or unfolding given in a Banach space may be replaced by a germ or unfolding in a finite-dimensional space in such a way that the geometry of the bifurcation set and the overlying critical point set is preserved. The aim is to provide a practicable tool, not an exhaustive theoretical discussion. How practicable it is may be seen from another paper (Magnus and Poston(7)) in which it is applied to a problem in elasticity; indeed, the present paper in part was originally the ‘machinery’ section of that paper. The second aspect, stability, is of more theoretical interest. The well known structural stability of universal unfoldings (see (5)) is extended to infinite-dimensions. Only a local theory is given.

Let A denote the mod 2 Steenrod algebra. Let ℤ2[x, x−l] be the (graded) ring of finite Laurent series over ℤ2 in the variable x with dim (x) = 1. ℤ2[x, x−1] is a module over the Steenrod algebra A by

where are binomial coefficients modulo 2 and m > 0 is large compared with |k| and i. Let M be the A-submodule of ℤ2[x, x−1 ] generated by all powers xi with i ≠ −1. It is easy to see that ℤ2 [,x, x−1]/M ≅ σ−1ℤ2 (means ℤ2 on dimension − 1). Let ρ: ℤ2[x, x−1] → σ−1 ℤ2 be the projection map.