Alexandre V. Borovik

Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus

Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Björling, an infinitesimal definition of the criterion of uniform convergence. Cauchy’s foundational stance is hereby reconsidered.

Finite and Locally Finite Groups

Preface. Introduction. Simple locally finite groups B. Hartley. Algebraic groups G.M. Seitz. Subgroups of simple algebraic groups and related finite and locally finite groups of Lie type M.W. Liebeck. Finite simple groups and permutation groups J. Saxl. Finitary linear groups: a survey R.E. Phillips. Locally finite simple groups of finitary linear transformations J.I. Hall. Non-finitary locally finite simple groups U. Meierfrankenfeld. Inert subgroups in simple locally finite groups V.V. Belyaev. Group rings of simple locally finite groups A.E. Zalesskii. Simple locally finite groups of finite Morley rank and odd type A.V. Borovik. Existentially closed groups in specific classes F. Leinen. Groups acting on polynomial algebras R.M. Bryant. Characters and sets of primes for solvable groups I.M. Isaacs. Character theory and length problems A. Turull. Finite p-groups A. Shalev. Index.

MULTIPLICATIVE MEASURES ON FREE GROUPS

We introduce a family of multiplicative distributions {μs|s∈(0,1)} on a free group F and study it as a whole. In this approach, the measure of a given set R⊆F is a function μ(R) : s → μs(R), rather then just a number. This allows one to evaluate sizes of sets using analytical properties of their measure functions μ(R). We suggest a new hierarchy of subsets R in F with respect to their size, which is based on linear approximations of the function μ(R). This hierarchy is quite sensitive, for example, it allows one to differentiate between sets with the same asymptotic density. Estimates of sizes of various subsets of F are given.

Symplectic Matroids

A symplectic matroid is a collection B of k-element subsets of J = {1, 2, ..., n, 1*, 2*, ...; n*}, each of which contains not both of i and i* for every i ≤ n, and which has the additional property that for any linear ordering ≺ of J such that i ≺ j implies j* ≺ i* and i ≺ j* implies j ≺ i* for all i, j ≤ n, B has a member which dominates element-wise every other member of B. Symplectic matroids are a special case of Coxeter matroids, namely the case where the Coxeter group is the hyperoctahedral group, the group of symmetries of the n-cube. In this paper we develop the basic properties of symplectic matroids in a largely self-contained and elementary fashion. Many of these results are analogous to results for ordinary matroids (which are Coxeter matroids for the symmetric group), yet most are not generalizable to arbitrary Coxeter matroids. For example, representable symplectic matroids arise from totally isotropic subspaces of a symplectic space very similarly to the way in which representable ordinary matroids arise from a subspace of a vector space. We also examine Lagrangian matroids, which are the special case of symplectic matroids where k = n, and which are equivalent to Bouchets symmetric matroids or 2-matroids.

Maximal subgroups in finite and profinite groups

We prove that if a finitely generated profinite group G is not generated with positive probability by finitely many random elements, then every finite group F is obtained as a quotient of an open subgroup of G. The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann. of Math. 137 (1993), 203-220, we then prove that a finite group G has at most IGIc maximal soluble subgroups, and show that this result is rather useful in various enumeration problems.

THE CONJUGACY PROBLEM IN AMALGAMATED PRODUCTS I: REGULAR ELEMENTS AND BLACK HOLES

We discuss the time complexity of the word and conjugacy problems for free products of two groups with amalgamation over a subgroup. We stratify the set of elements of the fre product with respect to the complexity of the word and conjugacy problems and show that for the generic stratum the conjugacy search problem is decidable under some reasonable assumptions about the groups groups involved. Moreover, the decision algorithm is fast on the generic stratum.

Coxeter matroid polytopes

If Δ is a polytope in real affine space, each edge of Δ determines a reflection in the perpendicular bisector of the edge. The exchange groupW (Δ) is the group generated by these reflections, and Δ is a (Coxeter) matroid polytope if this group is finite. This simple concept of matroid polytope turns out to be an equivalent way to define Coxeter matroids. The Gelfand-Serganova Theorem and the structure of the exchange group both give us information about the matroid polytope. We then specialize this information to the case of ordinary matroids; the matroid polytope by our definition in this case turns out to be a facet of the classical matroid polytope familiar to matroid theorists.

On the Schur-Zassenhaus Theorem for Groups of Finite Morley Rank

The Schur-Zassenhaus Theorem is one of the fundamental theorems of finite group theory. Here is its statement: Fact 1.1 ( Schur-Zassenhaus Theorem). Let G be a finite group and let N be a normal subgroup of G. Assume that the order ∣N∣ is relatively prime to the index [G:N]. Then N has a complement in G and any two complements of N are conjugate in G . The proof can be found in most standard books in group theory, e.g., in [S, Chapter 2, Theorem 8.10]. The original statement stipulated one of N or G/N to be solvable. Since then, the Feit-Thompson theorem [FT] has been proved and it forces either N or G/N to be solvable. (The analogous Feit-Thompson theorem for groups of finite Morley rank is a long standing open problem). The literal translation of the Schur-Zassenhaus theorem to the finite Morley rank context would state that in a group G of finite Morley rank a normal π-Hall subgroup (if it exists at all) has a complement and all the complements are conjugate to each other. (Recall that a group H is called a π-group , where π is a set of prime numbers, if elements of H have finite orders whose prime divisors are from π. Maximal π-subgroups of a group G are called π- Hall subgroups . They exist by Zorns lemma. Since a normal π-subgroup of G is in all the π-Hall subgroups, if a group has a normal π-Hall subgroup then this subgroup is unique.) The second assertion of the Schur-Zassenhaus theorem about the conjugacy of complements is false in general. As a counterexample, consider the multiplicative group ℂ* of the complex number field ℂ and consider the p -Sylow for any prime p , or even the torsion part of ℂ*. Let H be this subgroup. H has a complement, but this complement is found by Zorns Lemma (consider a maximal subgroup that intersects H trivially) and the use of Zorns Lemma is essential. In fact, by Zorns Lemma, any subgroup that has a trivial intersection with H can be extended to a complement of H . Since ℂ* is abelian, these complements cannot be conjugated to each other.

Model Theory with Applications to Algebra and Analysis: Permutation groups of finite Morley rank

The paper bounds the Morley rank of a definably primitive permutation group of finite Morley rank in terms of the rank of the set on which it acts.

THE ANDREWS–CURTIS CONJECTURE AND BLACK BOX GROUPS

The paper discusses the Andrews–Curtis graph Δk(G,N) of a normal subgroup N in a group G. The vertices of the graph are k-tuples of elements in N which generate N as a normal subgroup; two vertices are connected if one of them can be obtained from another by certain elementary transformations. This object appears naturally in the theory of black box finite groups and in the Andrews–Curtis conjecture in algebraic topology [3]. We suggest an approach to the Andrews–Curtis conjecture based on the study of Andrews–Curtis graphs of finite groups, discuss properties of Andrews–Curtis graphs of some classes of finite groups and results of computer experiments with generation of random elements of finite groups by random walks on their Andrews–Curtis graphs.