UNLABELLED Certain types of genetic analysis are simplified by assembling a collection of unrelated individuals, e.g. case-control experiments. If a family study is being performed then it will be necessary to extract subsets of unrelated, available individuals from pedigress. Our program provides an optimal method for performing this task. AVAILABILITY The software is available, free of char...

1 Preliminary definitions Definition 1 (S-Lenstra constant) Let K be a number field, and Z K its ring of integers. Let S be a finite set of valuations of K containing all the infinite valuations. Let L be a maximal subset of Z K such that x, y ∈ L ⇒ x − y ∈ Z if L = {a 0 , a 1 ,. .. , a k }. In fact we will use this second form of the condition. Remark 2 We can assume that if L is such a maxima...

A (d, k) set is a subset of R containing a translate of every k-dimensional disc of diameter 1. We show that if (1 + √ 2)k−1 + k > d and k ≥ 2, then every (d, k) set has positive Lebesgue measure. This improves a result of Bourgain, who showed that the analogous statement holds when 2k−1 +k ≥ d and k ≥ 2. We obtain this improvement in two parts. First, we replace Bourgain’s main estimate with a...

Let K be a field of characteristic two, and let λ be a two-part partition of some natural number r. Denote the permutation module corresponding to the (maximal) Young subgroup Σλ in Σr byM . We construct a full set of orthogonal primitive idempotents of the centraliser subalgebra SK(λ) = 1λSK(2, r)1λ = EndKΣr (M ) of the Schur algebra SK(2, r). These idempotents are naturally in one-to-one corr...

If P is a p-group for some prime p we shall write M (P ) to denote the set of all maximal subgroups of P and Md(P ) = {P1, ..., Pd} to denote any set of maximal subgroups of P such that ∩d i=1 Pi = Φ(P ) and d is as small as possible. In this paper, the structure of a finite group G under some assumptions on the c-normal or s-quasinormally embedded subgroups in Md(P ), for each prime p, and Syl...

Let G be a group, and S a non-empty subset of G. Then S is product-free if ab / ∈ S for all a, b ∈ S. We say S is locally maximal product-free if S is product-free and not properly contained in any other product-free set. A natural question is what is the smallest possible size of a locally maximal product-free set in G. The groups containing locally maximal product-free sets of sizes 1 and 2 w...

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G/U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct certain canonical G-invariant space S(X) (called the Schwartz space of X) of functions on X , which is an extension of the space of smooth compactly supported functions on X . We show that the s...

Algorithms are constructed which, when an explicit presentation of a finitely generated metabelian group G in the variety si1 is given, produce unitary presentations for the derived subgroup G', the centre Z{G), the Fitting subgroup Fit(G), and the Frattini subgroup <p(G). Additional algorithms of independent interest are developed for commutative algebra which construct the associated set of p...

Received 16 March 2008; accepted 4 May 2008 Abstract: We classify the maximal irreducible periodic subgroups of PGL(q,F) , where F is a field of positive characteristic p transcendental over its prime subfield, q 6= p is prime, and F× has an element of order q . That is, we construct a list of irreducible subgroups G of GL(q,F) containing the centre F×1q of GL(q,F) , such that G/F×1q is a maxim...

A basic fact of life Let G be a connected reductive group over Q. Set G∞ = G(R), and let K∞ ⊂ G∞ be a maximal compactmod-center subgroup, so D∞ = G∞/K∞ is the usual symmetric space for G. Following the notation in Borel and Wallach’s book, we set l0 = rank(G∞) − rank(K∞) and q0 = 1 2 (dimD∞ − l0). These are both nonnegative integers. For any open compact subgroup K ⊂ G(Af ) we have the usual lo...