For n a positive integer, a group G is called core-n if H/HG has order at most n for every subgroup H of G (where HG is the normal core of H, the largest normal subgroup of G contained in H). It is proved that a locally finite core-n group G has an abelian subgroup whose index in G is bounded in terms of n.

We determine all primitive groups which do not have a regular orbit on the power set of the permutation domain. As a corollary, we also determine all families of orbit equivalent primitive permutation groups.

1. Introduction

The classical circle method of Hardy and Littlewood (see Baker [1]) ensures that the diophantine equation

a1xk1+[ctdot ]+ asxks =0 (1.1)

has a nontrivial solution in nonnegative integers x1, [ctdot ], xs, provided s[ges ]C(k) and the coefficients a1, [ctdot ], as are all integers and not all the same sign.

Cappell conjectured the following in ]2]; compare ]11, Problem 2].

Conjecture 0.1. Let π be the fundamental group of a knot complement in [ ]3. Then Lhn(π)≃ Lhn(ℤ), for all n, where Lhnis the Wall–Novikov surgery group for homotopy equivalence, and the isomorphism is induced from the abelianization homomorphism π→ℤ.

Given a field k and a finite group G acting on the rational function field k(X1, . . ., Xn) as a group of k-automorphisms, an important Noether's problem asks whether the invariant subfieldformula here is purely transcendental over k.

Let G be a finite simple group. A conjecture of J. D. Dixon, which is now a theorem (see [2, 5, 9]), states that the probability that two randomly chosen elements x, y of G generate G tends to 1 as [mid ]G[mid ]→∞. Geoff Robinson asked whether the conclusion still holds if we require further that x, y are conjugate in G. In this note we study the probability Pc(G) that 〈x, xy〉=G, where x, y∈G are chosen at random (with uniform distribution on G×G). We shall show that Pc(G)→1 as [mid ]G[mid ]→∞ if G is an alternating group, or a projective special linear group, or a classical group of bounded dimension. In fact, some (but not all) of the exceptional groups of Lie type will also be dealt with.

We estimate the growth order in n of the Taylor coefficients of bounded univalent functions in the unit disk. Our estimate depends on the norm of the Grunsky operator. In particular, we improve the known results for coefficients of quasiconformally extendible univalent functions.

Let G be a finite group acting faithfully and transitively on a set Ω of size m, and let E = {x1, …, xk} be a generating set for G with x1x2…xk = 1. If x∈G has cycles of length r1, …, rl in its action on Ω, define ind (x) = [sum ]l1(ri−1). Then the genusg = g(G, Ω, E) is defined by

formula here

The descent algebra of the symmetric group, over a field of non-zero characteristic p, is studied. A homomorphism into the algebra of generalised p-modular characters of the symmetric group is defined. This is then used to determine the radical, and its nilpotency index. It also allows the irreducible representations of the descent algebra to be described.

In this paper we solve the embedding problem for partial transitive triple systems of order n and index λ, partial TTS (n, λ) s, showing that any partial TTS (n, λ) can be embedded in a TTS (v, λ) for all λ-admissible v[ges ]2n+1. This lower bound on v is the best possible. A simple proof is also given of a result of Colbourn and Harms which shows that every partial triple system of order n and index 2λ is the underlying triple system of a partial TTS (n, λ).

Necessary and sufficient conditions are obtained for a bipartite graph to admit a matching which includes a preassigned set of edges and excludes another preassigned set of edges.

A group is said to be (2,3)-generated if it can be generated by an element of order 2 and an element of order 3. The class of (2,3)-generated groups seems to be rather extensive. From results of Schupp [7] and Mason and Pride [3], it is known that there are 2ℵ0 isomorphism classes of (2,3)-generated groups, and moreover that every countable group can be embedded in a (2,3)-generated group. All finite non-abelian simple groups are (2,3)-generated, with the exception of some groups of low rank in characteristics 2 and 3 (see Wilson [13] or Sanchini and Tamburini [6]).

A set A of real numbers is called universal (in measure) if every measurable set of positive measure necessarily contains an affine copy of A. All finite sets are universal, but no infinite universal sets are known. Here we prove some results related to a conjecture of Erdős that there is no infinite universal set. For every infinite set A, there is a set E of positive measure such that (x+tA)⊆E fails for almost all (Lebesgue) pairs (x, t). Also, the exceptional set of pairs (x, t) (for which (x+tA)⊆E) can be taken to project to a null set on the t-axis. Finally, if the set A contains large subsets whose minimum gap is large (in a scale-invariant way), then there is E⊆R of positive measure which contains no affine copy of A.

We describe an uncountable family of compact group automorphisms with entropy log 2. Each member of the family has a distinct dynamical zeta function, and the members are parametrised by a probability space. A positive proportion of the members have positive upper growth rate of periodic points, and almost all of them have an irrational dynamical zeta function.

If infinitely many Mersenne numbers have a bounded number of prime divisors, then a typical member of the family has upper growth rate of periodic points equal to log 2, and lower growth rate equal to zero.

The present paper deals with real infinite-dimensional normed spaces; some of the main concepts here make sense, and have been treated in the literature, in the general context of topological Hausdorff linear spaces over reals.

A subset of a normed space X is a body if it is different from X itself and is the closure of its non-empty interior. A covering of X by bodies is called a tiling ofX whenever any two different members of it have disjoint interiors. The elements of such a covering are called tiles. A tiling is bounded (respectively convex) whenever each tile is bounded (respectively convex).

The norm of a group G is the subgroup of elements of G which normalise every subgroup of G. We shall denote it κ(G). An ascending series of subgroups κi(G) in G may be defined recursively by: κ0(G) = 1 and, for i[ges ]0, κi+1(G)/κi(G) = κ(G/κi(G)). For each i, the section κi+1(G)/κi(G) clearly contains the centre of the group G/κi(G). A result of Schenkman [8] gives a very close connection between this norm series and the upper central series: ζi(G)⊆κi(G) ⊆ζ2i(G).

Yang proved in [10] that if f and f(k) have no fix-points for every f∈F, where F is a family of meromorphic functions in a domain G and k a fixed integer, then F is normal in G. In this paper we prove normality for families F for which every f∈F omits ψ1 and f(k) omits ψ2, where ψ1 and ψ2 are analytic functions with ψ(k)1[nequiv ]ψ2.

Let V be a vector space over some division ring D, and G a finitary subgroup of GL(V). If G is locally completely reducible, then the D-G modules V, [V, G] and V/CV(G) need not be completely reducible, even if dimDV is finite. Moreover, if F is a field, then V and V/CV(G) need not be completely reducible. We prove here that if D is a finite-dimensional division algebra and G is locally completely reducible, then [V, G] is always a completely reducible D-G module.

Let Ω be a bounded connected open set in ℝN, N[ges ]2, and let −ΔΩ[ges ]0 be the Dirichlet Laplacian defined in L2(Ω). Let λΩ>0 be the smallest eigenvalue of −ΔΩ, and let ϕΩ>0 be its corresponding eigenfunction, normalized by ∥ϕΩ∥2=1. For sufficiently small ε>0 we let R(ε) be a connected open subset of Ω satisfying

formula here

Let −Δε[ges ]0 be the Dirichlet Laplacian on R(ε), and let λε>0 and ϕε>0 be its ground state eigenvalue and ground state eigenfunction, respectively, normalized by ∥ϕε∥2=1. For functions f defined on Ω, we let Sεf denote the restriction of f to R(ε). For functions g defined on R(ε), we let Tεg be the extension of g to Ω satisfying

formula here

Let G and A be finite groups with coprime orders, and suppose that A acts on G by automorphisms. Let π(G, A)[ratio ]IrrA(G)→Irr (CG(A)) be the Glauberman–Isaacs correspondence. Let B[les ]A and χ∈IrrA(G). We exhibit a counterexample to the conjecture that χπ(G, A) is an irreducible constituent of the restriction of χπ(G, B) to CG(A).