A version of the Jensen–Johnsbråten coding at arbitrary level n≥ 3

We generalize, on higher projective levels, a construction of “incompatible” generic Δ¹ ₃ real singletons given by Jensen and Johnsbr?ten. Received: 3 November 1998 / Revised version: 23 April 2000 / Published online: 3 October 2001     

2-step nilpotent Lie groups of higher rank

We construct a family of simply connected 2-step nilpotent Lie groups of higher rank such that every geodesic lies in a flat. These are as Riemannian manifolds irreducible and arise from real representations of compact Lie algebras. Moreover we show that groups of Heisenberg type do not even infinitesimally have higher rank. Received: 2 July 2001 / Revised version: 19 October 2001     

Maximal cofinitary groups

We discuss the cardinalities of maximal cofinitary groups under the assumption of . We also discuss various open questions in this area. Received: 24 July 1997     

On the weak Freese–Nation property of ?(ω)

Continuing [6], [8] and [16], we study the consequences of the weak Freese-Nation property of (?(ω),⊆). Under this assumption, we prove that most of the known cardinal invariants including all of those appearing in Cichoń's diagram take the same value as in the corresponding Cohen model. Using this principle we could also strengthen two results of W. Just about cardinal sequences of superatomic Boolean algebras in a Cohen model. These results show that the weak Freese-Nation property of (?(ω),⊆) captures many of the features of Cohen models and hence may be considered as a principle axiomatizing a good portion of the combinatorics available in Cohen models. Received: 7 June 1999 / Revised version: 17 October 1999 /?Published online: 15 June 2001     

The class of groups which have a subgroup of index 2 is not elementary

F. Oger proved that if A is a finite group, then the class of groups which are abelian-by-A can be axiomatized by a single first order sentence. It is established here that, in Oger's result, the word abelian cannot be replaced by group. Received: 15 March 1996 / Published online: 18 July 2001     

Groupwise dense families

We show that the Filter Dichotomy Principle implies that there are exactly four classes of ideals in the set of increasing functions from the natural numbers. We thus answer two open questions on consequences of ? < ?. We show that ? < ? implies that ? = ?, and that Filter Dichotomy together with ? < ? implies ? < ?. The technical means is the investigation of groupwise dense sets, ideals, filters and ultrafilters. With related techniques we prove the new inequality ?≤ cf(?). Received: 9 October 1998 / Revised version: 18 August 1999 / Published online: 21 December 2000     

Near coherence and filter games

We investigate a two-player game involving pairs of filters on ω. Our results generalize a result of Shelah ([7] Chapter VI) dealing with applications of game theory in the study of ultrafilters. Received: 28 September 1998 / Published online: 25 January 2001     

A Concentration Theorem for the Heat Equation

 We compare the solution of to the solution of the same equation where f is replaced by a “concentrated” source . As a result we derive some estimates on the solution in spatial norm, locally uniformly in t, with respect to the norm of for any integer . In the case we obtain a critical inequality relating the norm of to an exponential norm of u. (Received 1 September 2000; in revised form 17 January 2001)     

Recursive events in random sequences

Let ω be a Kolmogorov–Chaitin random sequence with ω_{1: n} denoting the first n digits of ω. Let P be a recursive predicate defined on all finite binary strings such that the Lebesgue measure of the set {ω|∃nP(ω_{1: n} )} is a computable real α. Roughly, P holds with computable probability for a random infinite sequence. Then there is an algorithm which on input indices for any such P and α finds an n such that P holds within the first n digits of ω or not in ω at all. We apply the result to the halting probability Ω and show that various generalizations of the result fail. Received: 1 December 1998 / Published online: 3 October 2001     

Cyclic regulating balanced Butler groups

A necessary and sufficient condition is derived for an arbitrary cyclic regulating almost completely decomposable group to belong to a K(n)-class, n≥ 0. This allows one to formulate a Structure Theorem for clipped p-primary cyclic regulating groups in each K(n)-class. Using the Indecomposability Criterion we construct an example of an indecomposable balanced Butler group which does not have a unique regulating subgroup. Received: 3 November 1998     

Generalized blocks for symmetric groups

We study, via character-theoretic methods, an ℓ-analogue of the modular representation theory of the symmetric group, for an arbitrary integer ℓ≥2. We find that many of the invariants of the usual block theory (ie. when ℓ is prime) generalize in a natural fashion to this new context. Oblatum 21-III-2002 & 5-VIII-2002?Published online: 8 November 2002     

Coleman automorphisms of finite groups

An automorphism of a finite group G whose restriction to any Sylow subgroup equals the restriction of some inner automorphism of G shall be called Coleman automorphism, named for D. B. Coleman, who's important observation from [2] especially shows that such automorphisms occur naturally in the study of the normalizer of G in the units of the integral group . Let Out be the image of these automorphisms in Out. We prove that Out is always an abelian group (based on previous work of E. C. Dade, who showed that Out is always nilpotent). We prove that if no composition factor of G has order p (a fixed prime), then Out is a -group. If O, it suffices to assume that no chief factor of G has order p. If G is solvable and no chief factor of has order 2, then , where is the center of . This improves an earlier result of S. Jackowski and Z. Marciniak. Received: 26 May 2000; in final form: 5 October 2000 / Published online: 19 October 2001     

A superatomic Boolean algebra with few automorphisms

Assuming GCH, we prove that for every successor cardinal μ > ω₁, there is a superatomic Boolean algebra B such that |B| = 2^μ and |Aut B| = μ. Under ◊_ω1, the same holds for μ = ω₁. This answers Monk's Question 80 in [Mo]. Received: 1 January 1998 / Revised version: 18 May 1999 / Published online: 21 December 2000     

Every incomplete computably enumerable truth-table degree is branching

If r is a reducibility between sets of numbers, a natural question to ask about the structure ? _r of the r-degrees containing computably enumerable sets is whether every element not equal to the greatest one is branching (i.e., the meet of two elements strictly above it). For the commonly studied reducibilities, the answer to this question is known except for the case of truth-table (tt) reducibility. In this paper, we answer the question in the tt case by showing that every tt-incomplete computably enumerable truth-table degree a is branching in ? _tt . The fact that every Turing-incomplete computably enumerable truth-table degree is branching has been known for some time. This fact can be shown using a technique of Ambos-Spies and, as noticed by Nies, also follows from a relativization of a result of Degtev. We give a proof here using the Ambos-Spies technique because it has not yet appeared in the literature. The proof uses an infinite injury argument. Our main result is the proof when a is Turing-complete but tt-incomplete. Here we are able to exploit the Turing-completeness of a in a novel way to give a finite injury proof. Received: 22 January 1999 / Revised version: 12 July 1999 / Published online: 21 December 2000     

Valuation-like maps and the congruence subgroup property

Let D be a finite dimensional division algebra and N a subgroup of finite index in D ^×. A valuation-like map on N is a homomorphism ϕ:N?Γ from N to a (not necessarily abelian) linearly ordered group Γ satisfying N _<-α+1⊆N _<-α for some nonnegative α∈Γ such that N _<-α≠=?, where N _<-α={x∈N|ϕ(x)<-α}. We show that this implies the existence of a nontrivial valuation v of D with respect to which N is (v-adically) open. We then show that if N is normal in D ^× and the diameter of the commuting graph of D ^×/N is ≥4, then N admits a valuation-like map. This has various implication; in particular it restricts the structure of finite quotients of D ^×. The notion of a valuation-like map is inspired by [27], and in fact is closely related to part (U3) of the U-Hypothesis in [27]. Oblatum 14-VII-2000 & 22-XI-2000?Published online: 5 March 2001     

Some results on supersolvability of finite groups

We get new characterizations of finite supersolvable groups based on the structure of the generalized Fitting subgroup. Then we extend our results to the saturated formations containing the class of all supersolvable groups. Authors’ addresses: M. Asaad, Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt; Piroska Cs?rg?, Department of Algebra and Number Theory, E?tv?s University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary     

Meeting infinitely many cells of a partition once

We investigate several versions of a cardinal characteristic defined by Frankiewicz. Vojtáš showed , and Blass showed . We show that all the versions coincide and that is greater than or equal to the splitting number. We prove the consistency of and of . Received: 2 October 1996 / Revised version: 22 May 1997     

Explaining the Gentzen–Takeuti reduction steps: a second-order system

Using the concept of notations for infinitary derivations we give an explanation of Takeuti's reduction steps on finite derivations (used in his consistency proof for Π¹ ₁-CA) in terms of the more perspicious infinitary approach from [BS88]. Received: 27 April 1999 / Published online: 21 March 2001