Character tables of p-groups with derived subgroup of prime order III

Two character tables of finite groups are isomorphic if there exist a bijection for the irreducible characters and a bijection for the conjugacy classes that preserve all the character values. In this paper we compare the number of non-isomorphic p-groups with derived subgroup of order p with the number of non-isomorphic character tables of these groups. We show that the difference between the number of non-isomorphic groups of this type and the number of non-isomorphic character tables of these groups increases exponentially. Furthermore, we prove that if we fix the index of the center of these groups, say $p^{2m}$, and we let the size of the groups grow bigger, then, for each character table there are on average $(2m)!/(2^{m}m!)$ non-isomorphic groups whose character table are isomorphic to the given one.     

Theory of path length distributions I

The Path Length Distribution (PLD) of a (p, q) graph is defined to be the array (X₀, X₁, X₂, …, X_p-1), where X₀ is the number of (unordered) pairs of vertices which have no path connecting them and X_l, 1 ≦ l ≦ p-1, is the number of pairs of vertices which are connected by a path of length l (see [1, 2]). The topic of this paper is the occurence of non-isomorphic graphs having the same path length distribution. For trees, a constructive procedure is given, showing that for any positive integer N there exist N non-isomorphic trees of diameter four which have the same PLD. Also considered are PLD-maximal graphs — those graphs with p vertices such that all pairs of vertices are connected by a path of length l for 2 ≦ l ≦ p-1. In addition to providing more examples of non-isomorphic graphs having the same PLD, PLD-maximal graphs are of intrinsic interest. For PLD-maximal graphs, we give sufficient degree and edge conditions and a necessary edge condition.     

A class of vertex-transitive digraphs,II

(1). We determine the number of non-isomorphic classes of self-complementary circulant digraphs with pq vertices, where p and q are distinct primes. The non-isomorphic classes of these circulant digraphs with pq vertices are enumerated. (2). We also determine the number of non-isomorphic classes of self-complementary, vertex-transitive digraphs with a prime number p vertices, and the number of self-complementary strongly vertex-transitive digraphs with p vertices. The non-isomorphic classes of strongly vertex-transitive digraphs with p vertices are also enumerated.     

A generalization of the Oberwolfach problem and Ct‐factorizations of complete equipartite graphs

We consider the following generalization of the Oberwolfach problem: “At a gathering there are n delegations each having m people. Is it possible to arrange a seating of mn people present at s round tables (where can accommodate people and ) for k different meals so that each person has every other person not in the same delegation for a neighbour exactly once?”. We will concentrate on the case when all tables accommodate the same number t of people, give a complete solution for t even and settle most cases for t odd. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 42–49, 2000     

Characterizing geometric designs, II

We provide a characterization of the classical point-line designs PG₁(n,q), where n?3, among all non-symmetric 2-(v,k,1)-designs as those with the maximal number of hyperplanes. As an application of this result, we characterize the classical quasi-symmetric designs PGn−2(n,q), where n?4, among all (not necessarily quasi-symmetric) designs with the same parameters as those having line size q+1 and all intersection numbers at least qn−4+?+q+1. Finally, we also give an explicit lower bound for the number of non-isomorphic designs having the same parameters as PG₁(n,q); in particular, we obtain a new proof for the known fact that this number grows exponentially for any fixed value of q.     

Counting finite index subgroups and the P. Hall enumeration principle

We apply the P. Hall enumeration principle to count the number of subgroups of a given index in the free pro-p group and the free abelian group. We shall present an infinite family of non-isomorphic pro-p groups with the same zeta function.     

Graphs with specified radius and diameter

The radius and diameter of a graph are known to satisfy the relation rad G ≤ diam G ≤ 2 rad G. We show that this is the only restriction on these parameters and construct all non-isomorphic graphs of minimal order having a specified radius and diameter.     

Invariant circles for homogeneous polynomial vector fields on the 2-dimensional sphere

LexX be a homogeneous polynomial vector field of degreen≥3 on S² having finitely many invariant circles. Then, for such a vector fieldX we find upper bounds for the number of invariant circles, invariant great circles, invariant circles intersecting at a same point and parallel circles with the same director vector. We give examples of homogeneous polynomial vector fields of degree 3 on S² having finitely many invariant circles which are not great circles, which are limit cycles, but are not great circles and invariant great circles that are limit cycles. Moreover, for the casen=3 we determine the maximum number of parallel invariant circles with the same director vector. The authors are partially supported by a MCYT grant BFM2002-04236-C02-02 and by a CIRIT grant number 2001SGR 00173.     

Zur Berechnung der Kapitulation in bikubischen Zahlkörpern

The capitulation kernel is the kernel of the natural extension homomorphism of the ideal class groups in a extension K|k of number fields. In this paper K is a non-cyclic Galois field of degree 6 over the rationals and k is its quadratic subfield. Two different methods of computing the capitulation kernel are discussed. Both depend on the relationship between capitulation and unit structure. The paper closes with two tables. They contain the capitulation kernel for all ramified extensions K|k having cubic discriminants between –20000 and 100000.     

Varieties without minimal generators

While many familiar varieties have a minimal varietal generator, i.e., a regular projective finitely presentable regular generator such that none of its retracts is a regular generator, and even a unique one, we present (a) a variety having no minimal varietal generator at all and (b) a variety having two non-isomorphic minimal varietal generators. Moreover we demonstrate that the same effects can happen with respect to a weaker notion of minimality and are common even in module categories. Received April 7, 1999; accepted in final form July 10, 2000.     

Indistinguishable Trees and Graphs

We show that a number of graph invariants are, even combined, insufficient to distinguish between non-isomorphic trees or general graphs. Among these are: the spectrum of eigenvalues (equivalently, the characteristic polynomial), the number of independent sets of all sizes or the number of connected subgraphs of all sizes. We therefore extend the classical theorem of Schwenk that almost every tree has a cospectral mate, and we provide an answer to a question of Jamison on average subtree orders of trees. The simple construction that we apply for this purpose is based on finding graphs with two distinguished vertices (called pseudosimilar) that do not belong to the same orbit but whose removal yields isomorphic graphs.     

The number of bipartite tournaments with a unique given factor

We give a recursive function in order to calculate the number of all non-isomorphic bipartite tournaments containing an unique hamiltonian cycle. Using this result we determine the number of all nonisomorphic bipartite tournaments that admit an unique factor isomorphic to a given 1-diregu-lar bipartite graph.     

Complete theories with finitely many countable models. II

Previously, we obtained a syntactic characterization for the class of complete theories with finitely many pairwise non-isomorphic countable models [1]. The most essential part of that characterization extends to Ehrenfeucht theories (i.e., those having finitely many (but more than 1) pairwise non-isomorphic countable models). As the basic parameters defining a finite number of countable models, Rudin-Keisler quasiorders are treated as well as distribution functions defining the number of limit models for equivalence classes w.r.t. these quasiorders. Here, we argue to state that all possible parameters given in the characterization theorem in [1] are realizable. Also, we describe Rudin-Keisler quasiorders in arbitrary small theories. The construction of models of Ehrenfeucht theories with which we come up in the paper is based on using powerful digraphs which, along with powerful types in Ehrenfeucht theories, always locally exist in saturated models of these theories. Supported by RFBR grant Nos. 02-01-00258 and 05-01-00411. __________ Translated from Algebra i Logika, Vol. 45, No. 3, pp. 314–353, May–June, 2006.     

On the distance between non-isomorphic groups

A result of Ben-Or, Coppersmith, Luby and Rubinfeld on testing whether a map between two groups is close to a homomorphism implies a tight lower bound on the distance between the multiplication tables of two non-isomorphic groups.     

Strongly inequivalent representations and Tutte polynomials of matroids

We develop constructive techniques to show that non-isomorphic 3-connected matroids that are representable over a fixed finite field and that have the same Tutte polynomial abound. In particular, for most prime powers q, we construct infinite families of sets of 3-connected matroids for which the matroids in a given set are non-isomorphic, are representable over GF(q), and have the same Tutte polynomial. Furthermore, the cardinalities of the sets of matroids in a given family grow exponentially as a function of rank, and there are many such families.In Memory of Gian-Carlo Rota     

The power graph of a finite group

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that non-isomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each order. We also show that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.     

Types directed by constants

Let T be a complete, countable, first-order theory having infinite models. We introduce types directed by constants, and prove that their presence in a model of T guaranties the maximal number of non-isomorphic countable models :  I(ℵ₀,T)=2^ℵ₀.     

Bemerkungen Zu Einer Klasse Kommutativer Inzidenzgruppen

We study a class of commutative incidence groups obtained by a special sort of couplings on affine translation planes over commutative quasifields. We give examples of different couplings on the same plane, leading to non-isomorphic incidence groups by generalizing a construction from Dickson. In some cases all thus obtained groups are algebraically isomorphic.     

How many quasiplatonic surfaces?

We show that the number of isomorphism classes of quasiplatonic Riemann surfaces of genus ≦ g has a growth of type The number of non-isomorphic regular dessins of genus ≦ g has the same growth type. Received: 8 February 2005; revised: 2 May 2005