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COMBINATORICS ABSTRACTS

CONFERENCE MATRICES AND SELFDUAL

TERNARY CODES

K.T. ARASU

Abstract

In this joint work with Aaron Gulliver, we investigate certain construction methods for selfdual ternary codes using a class of conference matrices.Examples are given for lengths up to 96. The codes of length 12, 24, 36 and 48 are equivalent to the Pless symmetry codes. In addition, we have determined that the minimum distance for the codes of length 60 and 84 are the same for both classes. The minimum distance calculations for length 96 (dimension 48) yields 24, improving the previously known bounds according to Brouwer's tables. For the case of length 84, we have determined that the minimum distance of the Pless symmetry code is exactly 21. The relationship between our codes and those of Pless (symmetry codes) is also examined.

EMBEDDABILITY IN DOWLING GEOMETRIES

PETER BROOKSBANK

Abstract

Recent results have revealed that Dowling geometries share a number of the nice properties of projective geometries; in some sense the former may be regarded as group-theoretic analogues of the latter. A fundamental result of Rado concerning projective geometries states that every finite geometry representable over a field is representable over a finite field. The corresponding question (posed by Bonin) for Dowling geometries is the following:

Is it true that a finite geometry that embeds in the Dowling geometry of an infinite group, necessarily embeds in a Dowling geometry of a finite group?

In this talk we demonstrate, constructively, that the above question has a negative answer. By way of commentary on the analogy between Dowling and projective geometries, we observe that although our findings contrast with Rado 's result for projective geometries, it is a purely group-theoretic phenomenon tha t settles the matter. This is joint work with Hongxun Qin, Edmund Robertson, and Ákos Seress.

THE $p'$-PART INVARIANT FACTORS

OF THE INCIDENCE MATRICES OF $r-1$-

AND $s-1$-DIMENSIONAL LINEAR SUBSPACES

IN $PG(n,q)$ WHEN $1<s\le r<n$

DAVID B. CHANDLER

Abstract

Let $V$ be an $(n+1)$-dimensional vector space over ${\rm GF}(q)$, where $q=p^t$, $p$ is a prime. For $1\le s\le r\le n$, let $A_{s,r}^n(q)$ be the (0,1)-incidence matrix with rows and columns respectively indexed by the $s$- and $r$-dimensional subspaces of $V$, and with $(X,Y)$-entry equal to one if and only if the $s$-dimensional subspace $X$ is contained in the $r$-dimensional subspace $Y$. The rank of $A_{s,r}^n(q)$ was computed by Yakir and Frumkin over any field not of characteristic $p$. In this talk, we extend their results to determine the $\ell$-adic Smith normal form of $A_{s,r}^n(q)$ for $\ell\ne p$.

< a name="davis1"> NEW AMORPHIC ASSOCIATION SCHEMES

JAMES A. DAVIS

Abstract

Applying results from quadratic forms, projective geometries, and recent results of Brouwer and van Dam, we construct the first known amorphic association scheme with negative Latin square type classes where the underlying group structure is not elementary abelian. We give a simple proof of a result of Hamilton that generalized Brouwer's result. We use multiple distinct quadratic forms to construct amorphic association schemes with a large number of classes.

This represents joint work with Qing Xiang.

SOME NEW CONSTRUCTIONS OF DIFFERENCE

MATRICES OVER $\mathbb Z_{3n}$

ANTHONY B. EVANS

Abstract

We will describe some new constructions of $(G, 5, 1)$-difference matrices, $G$ the cyclic group of order $3n$, $n$ divisible by neither 2 nor 3.

PATH-PERFECT BIPARTITE GRAPHS

PETER HAMBURGER

Abstract

A graph $G$ is path-perfect if there is a positive integer $n$ such that the edge set $E(G)$ of the graph $G$ can be partitioned into paths of length $1,2,3,\dots,n.$ We prove Fink and Strait's conjecture: A complete bipartite graph $K_{s,t}$ on $t+s$ vertices $(t\leq s)$ is path-perfect if and only if there is a positive integer $n$ such that the following two conditions are satisfied; (i) $n\le 2t,$ and (ii) $n(n+1)=2ts.$ This is a joint result with W. Cao.

SKEW POLYNOMIALS, SEMI-LINEAR

TRANSFORMATIONS AND THE NUMBER

OF NONEQUIVALENT CODES

XIANG-DONG HOU

Abstract

Two linear codes in ${\mathbb{F}}_q^n$ are called equivalent if one can be obtained from the other through the actions of a monomial matrix and an automorphism of ${\mathbb{F}}_q$. Let $N_{k,n}$ be the number of nonequivalent $k$-dimensional codes in ${\mathbb{F}}_q^n$. We describe a method for computing $N_{k,n}$. The method relies on the canonical forms of semi-linear transformations of ${\mathbb{F}}_q^n$, which are characterized by indecomposable skew polynomials over ${\mathbb{F}}_q$. Some numerical results will be presented.

AN INFINITE SERIES OF EXAMPLES OF

PROPER FINITE LOOPS HAVING A REGULAR

COLLINEATION GROUP

MIKHAIL KLIN

Abstract

Aiso Heinze, in his Ph.D Thesis (Oldenburg, Germany, 2001), has determined all partial difference sets (briefly pds's) over groups of order up to 49. In his research he was using the computer catalogues of strongly regular graphs with a small number of vertices (Ted Spence, University of Glasgow, Scotland),

Among his results was the proof of the existence of two unusual pds's over groups of order 36, one of these groups is a direct product of two copies of dihedral groups of order 6. Both these pds's imply the same (up to isomorphism) strongly regular graph $\Gamma$ with the parameters $v = 36$, $k = 15$, $\lambda = 6$. The automorphism group of $\Gamma$ has order 648, it acts transitively on the vertex set of $\Gamma$. Moreover, it turns out that $\Gamma$ is a Latin square graph, coming from a proper loop of order 6. (A loop is called proper if it does not contain a group in its main class.) Soon after we realized that A.Barlotti and K.Strambach were looking for exactly such example of a proper loop (Advances in Mathematics, 49:1-105, 1983).

We managed to get a quite beautiful computer free interpretation of our result. Moreover, starting from our construction, we were able to prove an existence of an infinite series of proper loops with similar properties. Our loops have order $2p$, where $p$ is a prime, equal to 3 modulo 4. Finally, we will discuss links between our results and various investigations of many other mathematicians, including: E. Schoenhardt, A. Sade, R.A. Bailey, C.E. Praeger, A. Sprague, E. Wilson, R.L. Wilson Jr., E.G. Goodaire & D.A. Robinson, K. Kunen.

In our research we were using computer packages COCO, GAP, GRAPE, nauty.

LINEAR CONNECTIVITY FORCES LARGE MINORS

JOHN MAHARRY

Abstract

Using Tree Decompositions and Robertson-Seymour's Excluded Minor Structure Theorem, we show that a linear (in terms of $a$) bound on the connectivity implies that sufficiently large graphs will contain $K_{a,k}$ and $K_a$ minors. In particular, for any $k$, any $7$-connected sufficiently large graph must contain a $K_{3,k}$-minor. Moreover, for any $k$, any $22a+1$-connected sufficiently large graph myst contain a $K_{a,k}$-minor. The conjectured connectivity is $2a+1$.

This is joint work with Thomas Boehme, Ken-ichi Kawarabayashi and Bojan Mohar.

SOME SPECIAL CASES OF HADWIGER'S CONJECTURE

ELIADE MICU

Abstract

Hadwiger's Conjecture states that the chromatic number of any graph is at most the size of the largest clique minor the graph possesses. It is believed that if a counterexample for the conjecture can be found, it should appear in the case of graphs with stability number equal to two, since they have large chromatic number. On the other hand, if the conjecture holds for this case, we obtain further proof that it should hold in general. In this talk, we establish the Hadwiger Conjecture for graphs with stability number two which do not contain induced cycles of size 4 or 5.

NEW FORMULAS FOR RAMANUJAN'S $\tau$ FUNCTION

AND OTHER CLASSICAL CUSP FORMS

STEPHEN C. MILNE

Abstract

Utilizing classical elliptic function invariants, we first sketch our derivation of several useful new formulas for Ramanujan's $\tau$ function. This work includes: the main pair of new formulas for the $\tau$ function that ``separate'' the two terms in the classical formula for the modular discriminant, a generating function form for both of these formulas, a Leech lattice form of one of these formulas, and a triangular numbers form. We then present analogous new formulas for several other classical cusp forms that appear in quadratic forms, sphere-packings, lattices and groups.

FORMALLY SELF-DUAL EVEN CODES OF LENGTH

DIVISIBLE BY 8

VERA PLESS

Abstract

A binary code with the same weight distribution as its dual is called formally self-dual(f.s.d.). We consider only even f.s.d. codes (all weights even). We show that if C is a Type l (not Type ll) s.d.[n,n/2,2[n/8]]code,with 8/n, then its weight distribution is unique. We determine the possible weight enumerators of a near-extremal f.s.d. even [n,n/2,2[n/8]] code with 8/n as well as the dimension of its radical.

This is joint work with Jon-Lark Kim.

RECONSTRUCTING TERNARY DOWLING GEOMETRIES

HONGXUN QIN

Abstract

We prove that the only geometry of rank $n>4$ all of whose proper contractions are ternary Dowling geometries is the ternary Dowling geometry. We use this to prove a stronger version of a conjecture of Joseph Kung and James Oxley.

This is joint work with Tom Dowling.

VORTEX STRUCTURE OF GRAPHS

NEIL ROBERTSON

Abstract

Vortex structure is a fundamental feature of graph minor structure theory. Perhaps it is also the least understood by the majority of graph theorists. This lecture will be an explanation of what is a vortex and how vortices fit into the structure theory and well-quasi-ordering aspects of graph minor theory.

$K$-SETS, CONVEX QUADRILATERALS, AND

SYLVESTER'S FOUR POINTS PROBLEM

GELASIO SALAZAR

Abstract

Back in 1865, Sylvester posed the following fundamental (and still open) problem in Geometric Probability. For a probability distribution $\mu$ in the plane, let $\square({\mu})$ denote the probability that four independent $\mu$-random points form a convex quadrilateral. What is Sylvester Four Point Problem's Constant $q_* : = \inf_{\mu} \square{\mu}$? (where the infimum is taken over all probability distributions with positive Borel measure). It is known that the problem of finding $q_*$ is closely related to the problem of determining the minimum number of convex quadrilaterals in a finite set of points in the plane in general position. This problem is in turn closely related to the rectilinear crossing number of the complete graphs $K_n$, and also to the problem of determining the number of $(\le k)$-sets in a finite set of points in the plane in general position. Recently, we have refined the best lower bound for $q_*$, and have shown that $0.37557 \le q_* \le 0.3802$. This is joint work with József Balogh.

< a name="seress1"> NONTRIVIAL $t$-INTERSECTION IN THE FUNCTION LATTICE

ÁKOS SERESS

Abstract

The function lattice, or generalized Boolean algebra, is the set of $\ell$-tuples with the $i$th coordinate an integer between 0 and a bound $n_i$. Two $\ell$-tuples $t$-intersect if they have at least $t$ common nonzero coordinates. We prove a Hilton-Milner type theorem for systems of $t$-intersecting $\ell$-tuples.

This is joint work with P. L. Erdos and L. Székely.

SIGNED-GRAPHIC MATROIDS

DANIEL C. SLILATY

Abstract

A signed graph is a pair $(G,f)$ in which $G$ is a graph and $f$ is a labeling of the edges with elements of the multiplicative group $\left \{+1,-1\right \}$. A circle in a signed graph is called negative if the product of labels on its edges is negative. Otherwise the circle is called positive. A signed graph is called balanced if all its circles are positive.

There is a matroid associated with $(G,f)$ whose elements are the edges of $G$. The rank of a subset $X$ of edges is equal to the number of vertices incident to edges in $X$ minus the number of balanced components in the subgraph defined by the $X$.

In this talk we will discuss several structural properties of this signed-graphic matroid: representability, topology, connectivity, and decompositions.

This talk presents joint work with Hongxun Qin of The George Washington university.

COMPUTING FREQUENT ITEMSETS

ALAN SPRAGUE

Abstract

The problem of generating frequent itemsets arises in the field of Data Mining. In this problem we are given a data file that may be regarded as a collection of subsets of some finite universal set. Each member of the universal set is called an "item". A "frequent itemset" is a set of items that appears at least $t$ times in the data file (for some threshold $t$). An important problem in Data Mining is to compute all frequent itemsets, and numerous algorithms for this have been given.

These algorithms are usually evaluated empirically - their relative execution time on various data files is reported. Analysis by computational complexity does not reach much beyond the observation that the worst case execution time is exponential (at least), since output size is exponential in terms of input size. We describe a variant of one well known algorithm to compute frequent itemsets, and sketch a proof that, when $t=1$, its execution time is linear in terms of the sum of output size and input size.

ON A SIMPLE GROUP OF ORDER 504, ITS

AUTOMORPHISM GROUP, AND RELATED

COMBINATORIAL STRUCTURES

ANDREW WOLDAR

Abstract

We consider the simple group $M=PSL(2,8)$ and its automorphism group $N=P\Gamma L(2,8)$ each in its classical action on 9 points. As these actions are known to be non-geometrical in the sense of D.Betten, we introduce a new notion, ``geometrical group of second order," which allows us to interpret our groups $M$ and $N$ of degree 9 (as well as their corresponding stabilzers of degree 8) as automorphism groups of suitable collections of regular uniform incidence structures.

Along the way we are led to consider so-called overlarge sets of Steiner designs, in particular the overlarge set of nine Steiner 3-designs with parameters $(8,4,1)$ which was initially found by W.L. Edge and later rediscovered by D.R. Breach and A.P. Street. From here, we give a simple new construction of a classical partial geometry with parameters $(K,R,T) = (8,9,4)$ having 120 points, 135 lines and automorphism group $A_9$. We also provide new constructions of the two partial geometries with parameters $(K,R,T)=(5,7,3)$ which were characterized by R. Mathon with the aid of a computer.

This is joint work with M. Klin and S. Reichard.

ON THE SMITH NORMAL FORM OF DESIGNS

QING XIANG

Abstract

Smith normal form is a generalization of $p$-rank. We use $p$-adic Jacobi sums, a theorem of Daqing Wan, and some representation of the general linear group to determine the Smith normal forms of designs arising in finite Desarguesian geometries.

GROUP THEORY ABSTRACTS

PRODUCTS OF CHARACTERS AND FINITE $p$-GROUPS

EDITH BANTE, University of Southern Mississippi Gulf Coast

Abstract

Let $p$ be an odd prime. Let $G$ be a finite $p$-group and $\chi$ be an irreducible character of $G$. Denote by $\overline{\chi}$ the complex conjugate of $\chi$. Assume that $\chi(1)=p^n$. We show that the number of distinct irreducible constituents of the product $\chi\overline{\chi}$ is at least $2n(p-1)+1$.

TOTALLY PERMUTABLE PRODUCTS OF

CERTAIN CLASSES OF FINITE GROUPS

J. BEIDLEMAN, Kentucky

Abstract

Subgroups $A$ and $B$ of a finite group $G$ are said to be totally permutable if every subgroup of $A$ permutes with every subgroup of $B$. The behaviour of finite pairwise totally permutable products are studied with respect to certain classes of groups including the class of groups where all the subnormal subgroups permute with all the maximal subgroups,the so called $SM$-groups and also the class of all groups where all the subnormal subgroups are permutable,the so called $PT$-groups (joint work with Peter Hauck and Hermann Heineken).

SOME CENTER-LIKE SUBSETS OF

GROUPS

HOWARD E. BELL, Brock University

Abstract

For a fixed positive integer $n$, define the Neumann $n$-center of the group $G$ to be the set of all elements $x$ of $G$ such that $xS = Sx$ for all $n$-subsets $S$ of $G$. Define the Freiman center to be the set of all $x$ in $G$ such that for each $y$ in $G$, the set $\{ xx, xy, yx, yy\}$ has at most three distinct elements. We discuss two questions:

1. When do these sets coincide with the center?

2. What are the other possibilities?

We also mention some analogues for rings.

FITTING FUNCTOR VALUES ON

DIRECT PRODUCTS

BEN BREWSTER, SUNY at Binghamton

A GROUP THEORY PROBLEM

ARISING FROM FINITE

GEOMETRIES

PETER BROOKSBANK, The Ohio State University

Abstract

Dowling geometries are a class of finite geometries defined using groups, which share many of the nice properties of projective geometries. A fundamental result of Rado concerning the latter type of geometries states that every finite geometry representable over a field is representable over a finite field. The corresponding question for Dowling geometries is the following:

Is it true that a finite geometry that embeds in the Dowling geometry for an infinite group, necessarily embeds in a Dowling geometry for a finite group?

This question can be reduced to a purely group-theoretic one, which may be stated roughly as follows:

Is it true that every finite partial Cayley table of an infinite group arises as a partial Cayley table of a finite group?

It is demonstrated that this latter question has a negative answer, thereby settling the original embeddability question. This is joint work with Honqxun Qin, Edmund Robertson, and Akos Seress.

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THE COMPLEX PLUS/MINUS

GEOMETRY

JON DUNLAP, Bowling Green

Abstract

Given an $(n+1)$-dimensional vector space $V$ over the field of complex numbers and a nondegenerate hermitian form, we consider the set of subspaces of $V$ that are either positive-definite or have a further positive-definite subspace of maximal dimension. We define a rank-n geometry on this set (using symmetrized containment as our incidence relation), demonstrate the simple-connectedness of this geometry in most cases, and pose some questions about the automorphism group of this geometry.

THE EXISTENCE PROBLEM FOR

COMPLETE MAPPINGS OF FINITE GROUPS

ANTHONY B. EVANS, Wright State University

Abstract

For $G$ a group, a mapping $\theta: G \to G$ is called a complete mapping of $G$ if the mappings $g \mapsto \theta(g)$ and $g \mapsto g\theta(g)$ are both bijections.

We derive a sufficient condition for a subgroup of $GL(n, F)$, $F$ the algebraic closure of $GF(2)$, to admit complete mappings. We will use this condition to give new proofs of the existence of complete mappings for some classes of linear groups.

COMPLEMENTARY AND SUPPLEMENTARY

SUBGROUP PROPERTIES IN FINITE

GROUPS

ARNOLD FELDMAN, Franklin and Marshall College

Abstract

We say a subgroup property $\alpha$ is closed under joins if $U_1 \alpha G$ and $U_2 \alpha G$ implies that $<U_1,U_2> \alpha G$. Then if $\alpha$ is closed under joins, given any subgroup $H$ of a group $G$, the $\alpha$ core of $H$ in $G$, $H_{\alpha G}$, can be defined to be $<U \vert U\le H$ and $U \alpha G>$, the unique maximal subgroup of $H$ that has property $\alpha$ in $G$. Note that if $U \alpha G$ means $U$ is normal in $G$, then $H_{\alpha G} = H_G$, the normal. We say a subgroup property $\beta$ is closed under intersections if $U_1 \beta G$ and $U_2 \beta G$ implies that $(U_1 \cap U_2) \beta G$. Then if $\beta$ is closed under intersections, given any subgroup $H$ of $G$, the $\beta$ closure of $H$ in $G$, $H^{\beta G}$, can be defined to be the intersection of all subgroups $U$ of $G$ such that $U \ge H$ and $U\beta G$; this is the unique minimal subgroup of $G$ that contains $H$ and has property $\beta$ in $G$. Note that if $U\beta G$ means $U$ is normal in $G$, then $H^{\beta G} = H^G$, the normal. We say a subgroup property $\gamma$ is a lattice property if it is closed under joins and closed under intersections, so that $\{H \in G \vert H \gamma G\}$ i Theorem 1. Suppose $\alpha$ is closed under joins, $\beta$ is closed under intersections, and $\gamma$ is a lattice property. Suppose further that if $U\gamma G$, then $U \alpha G$ and $U\beta G$. Then the following are equivalent: When (a) or (b) holds, so that both hold, we will say $\alpha$ and $\beta$ are $\gamma$-supplementary. We look at a variety of aspects of this concept, including, given a join-closed $\alpha$, the existence and determination of the unique weakest intersection-closed property, denoted $f_{\gamma}(\alpha)$, such that $\alpha$ and $f_{\gamma}(\alpha)$ are $\gamma$-supplementary. We say $\alpha$ and $f_{\gamma}(\alpha)$, are $\gamma$-complementary. Similarly, given an intersection-closed $\beta$, we determine the unique weakest join-closed property, $g_{\gamma}(\beta)$, such that $g_{\gamma}(\beta)$ and $\beta$ are $\gamma$ Theorem 2. Suppose $\alpha$ is in the image of $g_{\gamma}$ and $\beta$ is in the image of $f_{\gamma}$. Then $\beta = f_{\gamma}(\alpha)$ if and only if $\alpha = g_{\gamma}(\beta)$.

VIRTUAL HAMILTONIAN GROUPS

TUVAL FOGUEL, Auburn University - Montgomery

Abstract

In this talk we investigate the following problem: properties $P$ which transfer (or do not transfer) from all cyclic subgroups, or all abelian subgroups to all arbitrary subgroups. We solve this problem completely when $P$ is the property of having finite index in its normal closure, proving that $P$ carries from abelian, but not from cyclic to arbitrary subgroups. We use primarily some results of B.H. Neumann.

ON THE EXISTENCE OF FINITELY PRESENTED

INFINITE PERIODIC GROUPS AND

A QUESTION OF VERENA HUBA DYSON

ANTHONY GAGLION, US Naval Academy Annapolis MD

Abstract

This talk does not settle the issue of the existence of such groups. Assuming a first-order language, $L$, appropriate for group theory, the universal theory of a class of groups is just the set of all universal sentences of $L$ true for every group in the class. Sometime ago Verena Huber Dyson asked whether or not the universal theory of torsion groups coincides with the universal theory of finite groups. Here we show that if these theories coincide, then there cannot exist a finitely presented group of prime exponent. We cannot, however, say anything about the converse.

A METABELIAN GROUP HAVING ALL

CLASS 2 SUBGROUPS 2-SUBNORMAL

DAVID GARRISON, Lockheed Martin in Owego NY

Abstract

A subgroup $H$ of a group $G$ is called 2-subnormal, or subnormal of defect 2, if there exists a normal subgroup $K$ such that $H$ is normal in $K$. Heineken, Stadelmann, and Mahdavi have extensively studied groups having certain collections of subgroups subnormal of defect 2. Mahdavi looked at the interdependencies between groups having all cyclic, all abelian, all class 2, and all subgroups 2 subnormal. He left as an open question whether an arbitrary group having all class 2 subgroups 2-subnormal implies that all subgroups are 2-subnormal. We construct a group having all class 2 subgroups 2-subnormal, but not having all subgroups 2-subnormal.

CENTER AND NORM

H. HEINEKEN, Wuerzburg/Germany

Abstract

We establish a correlation between the structures of a group of power automorphisms of some group and their mutual commutator subgroup and consider the consequences for the norm of a group, and for its capability. (Report on work done by J.Beidleman, H.Heineken and M.Newell)

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ON GENERATING CLASSICAL GROUPS

C. HOFFMAN, Bowling Green

Abstract

This is joint work (in progress) with R Guralnick. We will prove that if $x$ is a nonparabolic element in a classical simple group then 3 conjugates of $x$ generate $G$. The methods used will be probabilistic in nature relying on Aschbacher characterization of maximal subgroups of classical groups.

CONJUGACY FUZZY MAPS

BETWEEN GROUPS

KENNETH JOHNSON, Penn State Abington

Abstract

The study of 2-characters of a group by Sehgal and the presenter led to the introduction of weak Cayley table maps (WCT maps) between groups.

A map $\alpha :G\rightarrow H$ is a WCT-homomorphism if

$$\left [ \alpha (g_{1})\alpha (g_{2})=\alpha (g_{1}g_{2})^{x} \right ]$$

where $x\in H$. A WCT-isomorphism is defined in the obvious manner.

The group of WCT isomorphisms from a group to itself has been studied by Humphries, who proved that for symmetric groups and free groups it is generated by the automorphisms and the map $x\mapsto x\symbol{94}-1$. A WCT-isomorphism between two groups ensures that they have the same character table and other properties.

We call a map $\alpha :G\rightarrow H$ is a conjugacy-fuzzy (cf)-homomorphism if

$$\left [ \alpha (g_{1})^{x_{1}}\alpha (g_{2})^{x_{2}}=\alpha (g_{1}g_{2}) : \right ]$$

where $x_{1},x_{2}\in H$.

The weaker property that defines fc maps has produced some surprising results. The cf-isomorphisms need not preserve conjugacy classes and can exist between groups with different character tables. Any permutation which fixes the conjugacy classes of a group setwise is a cf-isomorphism. If we consider the cf-isomorphisms from $G$ to itself there are three kinds of elements.

(1) Permutations which fix each conjugacy class.We call such permutations basic standard cf isomorphisms.

(2) cf-isomorphisms which permute conjugacy classes but do not split them.

We call the subgroup of permutations generated by cf-isomorphisms of types (1) and (2) the standard group.

(3) There are non-standard cf isos.

The set of all cf-isomorphisms is not necessarily a group but is a union of cosets of the standard group. We can characterize cf-isomorphisms by the property

$$\left [ a\text{ is a cf-isomorphism if and only if }\alpha (C_{i})\alpha (C_{j})=\alpha (C_{i}C_{j}) \right ]$$

for all pairs $($ $C_{i},C_{j})$ of conjugacy classes of $G$.

We do not, however, have a more intrinsic group theoretical or character theoretical characterization.

Many questions can be posed on the lines of: which groups have only standard cf maps, do free groups have non-standard maps, what equivalence relation does ``having a cf-isomorphism between them'' produce on the set of $p$-groups.

IN SEARCH OF THE SMALLEST GROUP,

WHERE FOR GIVEN n THE n $^{\text{\small TH}}$ POWERS OF

ELEMENTS DO NOT FORM A SUBGROUP

LUISE-CHARLOTTE KAPPE, SUNY at Binghamton

Abstract

It is well known that the squares of elements in a group do not form a subgroup and that the alternating group on four letters is minimal with this property. For given $n$, what is the group of minimal order such that the $n^{\text{\tiny th}}$ powers of elements do not form a subroup? For odd $n$, it can be shown that the dihedral group of order $2p$ is a minimal counter-examle, where $p$ is the smallest prime dividing $n$.

If $n$ is even, the situation is more complex. The order of the minimal counter example depends on the odd prime factors of $n$ and the exact $2$-power dividing $n$.

MODULARITY AND PERMUTABILITY IN THE

LATTICE OF $\Sigma$-PERMUTABLE SUBGROUPS

TOM KIMBER, SUNY Morrisville

Abstract

Let $\Sigma$ be a Hall system of a finite, solvable group $G$ and let $\mathcal P$$(\Sigma )$ denote the set of all $\Sigma$-permutable subgroups of $G$. It is known that $\mathcal P$$(\Sigma )$ is a sublattice of the subgroup lattice of $G$. We call $G$ an $SPM$-group if the lattice $\mathcal P$$(\Sigma )$ is modular. (The conjugacy of any two Hall systems implies that the condition $G \in SPM$ is independent of the Hall system chosen).

It is evident that if $U \perp V$ for all $U$, $V \in \mathcal P$$(\Sigma )$, then $\mathcal P$$(\Sigma )$ is modular. Our principal result is that the converse holds: If $G \in SPM$, then any two $\Sigma$-permutable subgroups permute with each other. In addition to the proof of this result, we will discuss closure properties of the class $SPM$ and local-global results (about what happens at each prime).

CHARACTER DEGREE GRAPHS OF

SOLVABLE GROUPS OF FITTING HEIGHT 2

MARK LEWIS, Kent State

Abstract

Let $G$ be a finite group, and let $cd(G)$ be the character degrees of $G$. We consider the graph whose vertex set is the set of primes that divide degrees in $cd(G)$. There is an edge between $p$ and $q$ if $pq$ divides some degree in $cd(G)$. These graphs have been studied in a number of places. In this paper, we classify the graphs that can arise when $G$ is a solvable group of Fitting height 2.

STRONGLY CONNECTIVE SUBSETS OF

CHARACTER DEGREE SETS

JOHN MCVEY, Kent State

Abstract

When $G$ is a finite nonabelian group, we associate the common-divisor graph ${\bf\Gamma}(G)$ with $G$ by letting nontrivial degrees in ${\rm cd}(G)$ be the vertices and making distinct vertices adjacent if they have a common nontrivial divisor. A set $\cal C$ of vertices for this graph is said to be strongly connective for ${\rm cd}(G)$ if there is some prime which divides every member of $\cal C$, and every vertex outside of $\cal C$ is adjacent to some member of $\cal C$. When $G$ has a nonabelian solvable quotient, we show that if ${\bf\Gamma}(G)$ is connected and has a diameter of at most $2$, then indeed ${\rm cd}(G)$ has a strongly connective subset.

ON COMMUTATORS IN $p$-GROUPS

ROBERT MORSE, Evansville Indiana

Abstract

For a given prime $p$, what is the smallest integer $n$ such that there exists a group of order $p^n$ in which the commutators are not a subgroup? In this paper we show that $n=6$ for any odd prime and $n=7$ for $p=2$ (Joint work with Luise-Charlotte Kappe).

CONJUGACY OF BN-INJECTORS

IN FINITE GROUPS

ANNI NEUMANN, Universität Tübingen

GROUP THEORETIC ORIGIN OF CERTAIN

GENERATING RELATIONS OF THE

HERMITE MATRIX FUNCTIONS

M.A. PATHAN, Aligarh Muslim University, Aligarh, India

Abstract

Weisner's group theoretic method is applied to obtain new generating functions for Hermite matrix functions $Tn(x,A)$ defined by L.Jodar and R.Com-pany, by giving suitable interpretation to the index $n$. The algebraic structure underlying Hermite matrix functions(HMF) can be recognized in full analogy with Hermite functions(HF), thus providing a unifying view to the theory of both HMF's and HF's. A few special cases of interest are also discussed.

CAP-PERSISTENCE AND

CAP-TRANSITIVITY:

A PRELIMINARY REPORT

JOSEPH PETRILLO, Franklin & Marshall College

Abstract

A subgroup $U$ of a finite group $G$ is said to have the cover-avoidance property in $G$ and is called a CAP-subgroup of $G$ if $U$ either covers or avoids each chief factor of $G$. It is not difficult to show that the cover-avoidance property is neither persistent nor transitive, but various properties of CAP-subgroups can be obtained by considering weaker forms of persistence and transitivity. In particular, for a CAP-subgroup $U$ of $G$, we say that $U$ is CAP-transitive in $G$ if each CAP-subgroup of $U$ is a CAP-subgroup of $G$, and we say that $U$ is CAP-persistent in $G$ if each CAP-subgroup of $G$ contained in $U$ is a CAP-subgroup of $U$.

GENERALIZATIONS OF $\cal T$-GROUPS

MATTHEW RAGLAND, Kentucky

Abstract

Let $\cal T$-groups, $\cal PT$-groups, and $\cal PST$-groups be those groups in which, respectively, normality, permutability, and Sylow-permu-tability are transitive. We say $G$ is a ${\cal T}_o$-group if $G/\Phi(G)$ is a ${\cal T}$-group, while $G$ is a Hall-$\cal X$ group if $G$ contains a normal nilpotent subgroup $N$ such that $G/N^{\prime}$ is an $\cal X$-group. Some basic results on ${\cal T}_o$-groups, Hall-$\cal T$ groups, Hall-$\cal PT$ groups, and Hall-$\cal PST$ groups will be discussed along with some characterization theorems (in the finite solvable case). A key step in characterizing the above groups is showing that they have a nilpotent hypercommutator which is also a Hall subgroup. Also key is the result that $G/\Phi(G)$ being a $\cal PST$-group implies $G/\Phi(G)$ is in fact a $\cal T$-group.

COMPUTING THE NONABELIAN TENSOR

SQUARE OF FREE NILPOTENT GROUPS

OF CLASS 3

JOANNE REDDEN, Evansville

Abstract

Preliminary report.

Generalizing a result for finite groups from Graham Ellis and Frank Leonard (1995) to polycylcic groups, we are able to apply both hand and computer methods to compute the nonabelian tensor square of the free nilpotent group of class 3 and rank $n$. Using this approach we will also reproduce results from our paper on the nonabelian tensor square of 2-Engel groups of rank $n$ to appear in Proceedings of the Edinburgh Mathematical Society (joint work with Russell D. Blyth and Robert F. Morse).

ASCENDING CENTRAL SERIES OF

WREATH PRODUCT $p$-GROUPS

JEFF RIEDL, University of Akron

Abstract

Fix any prime $p$ and positive integer $e$. Let $W(1,e)$ denote the cyclic group of order $p^e$. For each integer $n>1$, define $W(n,e)$ to be the regular wreath product group $W(n-1,e)\wr Z_p$. We mention that $W(n,1)$ is isomorphic to a Sylow $p$-subgroup of the symmetric group $Sym(p^n)$, and that for each prime power $q>1$ such that $p^e$ is the full $p$-part of $q-1$, the group $W(n,e)$ is isomorphic to a Sylow $p$-subgroup of $GL(p^{n-1},q)$ if and only if $p^e>2$.

We determine the ascending central series (and in the process, the nilpotence class) of $W(n,e)$, which has a very nice description. For $n>1$, its nilpotence class is $[e(p-1)+1]p^{n-2}$. Using similar methods, we also determine the ascending central series and the nilpotence class of the regular wreath product group $Z_{p^e}\wr E(m)$, where $E(m)$ is elementary abelian of order $p^m$, for an arbitrary positive integer $m$. We determine that this last group has nilpotence class $(e+m-1)(p-1)+1$.

A PHAN-TYPE THEOREM FOR

ORTHOGONAL GROUPS

ADAM ROBERTS, Bowling Green State University

Abstract

We will define a geometry on which an orthogonal group $G$ acts. We will show that this geometry is simply connected and using Tits' lemma we will be able to identify $G$ with the universal completion of an interesting amalgam of its subgroups.

GROUPS IN WHICH THE NORMALIZERS

FORM A CHAIN

JOE SMITH, SUNY Binghamton

RESIDUAL FINITENESS OF OUTER AUTOMORPHISMS OF

CERTAIN GENERALIZED FREE PRODUCTS

F.C.Y. TANG, Ottawa

Abstract

Outer automorphism groups of fundamental groups of surfaces of genus $k$ are known to be residually finite. These groups can be considered either as generalized free products or 1-relator groups. It is natural to ask which generalized free products or 1-relator groups have residually finite outer automorphism groups. We prove certain generalized free products have Property $E$, from which we derived that certain tree products of groups have residually finite outer automorphism groups. We apply these results to groups of linkages, hose knots and 1-relator groups with non-trivial centers (joint work with G. Kim).

ON SELF-NORMALITY AND ABNORMALITY

IN THE ALTERNATING GROUPS

MICHAEL WARD, Western Oregon University

Abstract

It is known that in a finite solvable group $G$, a subgroup $H$ is abnormal if and only if every subgroup of $G$ containing $H$ is self-normalizing in $G$. Although, in general, the assumption of solvability cannot be dropped, in this talk we discuss a proof of the theorem for the special case $G = A_n$ and $H$ a second maximal intransitive subgroup of $A_n$. Qinhai Zhang has previously spoken on this topic at these meetings in the case where $n$ is a power of a prime (joint work with Ben Brewster, Binghamton University and Qinhai Zhang, Shanxi Teachers University).

$p$-GROUPS WITH REGULAR

POWER STRUCTURE

LARRY WILSON, U of Florida

Abstract

Professor Kappe has studied the power margin in groups. One goal of this research is to find conditions on groups which force the powers of elements to behave in an aesthetically pleasing way. For $p$-groups, these three properties of regular $p$-groups are generally considered desirable:

1. $\{g^{p^i} : g \in G\}$ is a subgroup; that is, if $g$ and $h$ are in $G$ then $g^{p^i} \cdot h^{p^i} = x^{p^i}$ for some $x$ in $G$.
2. $\{g\in G: g^{p^i} = 1\}$ is a subgroup: that is, if $g^{p^i} = 1 = h^{p^i}$ then $(gh)^{p^i} = 1$.
3. The product of the orders of these subgroups is the order of the whole group.

We look at classes of $p$-groups known to satisfy these properties and discuss some related questions.

RING THEORY

SEMIPRIME $CS$ GROUP ALGEBRA OF

POLYCYCLIC-BY-FINITE GROUP WITHOUT

DOMAINS AS SUMMANDS IS HEREDITARY

ADEL N. ALAHMADI, Ohio University

Abstract

Antonio Behn proved that if $K[G]$ is prime $CS-$ring with $G$ polycyclic-by-finite, then $G$ is torsion-free or $G\cong D_{\infty }$ and $%% char(K)\neq 2$. In this paper we prove that if $K[G]$ is semiprime $CS-$ring having no domains summands with $G$ polycyclic-by-finite, then $K[G]$ is hereditary. Description of such group algebra, when $K$ is algebraically closed, is also given (JoinT work with S. K. Jain and J. B. Srivastava).

TORSION THEORETIC DUAL KRULL DIMENSION

VIA SUBDIRECT IRREDUCIBILITY

TOMA ALBU, Koc University, Istanbul, Turkey

Abstract

A right $R$-module $M$ is said to be subdirectly irreducible if it is nonzero and has a smallest nonzero submodule, or equivalently, if $M$ has a simple essential socle. Since this concept is dual to that of cyclic module, subdirectly irreducible modules are also called cocyclic. A submodule $N$ of $M$ is said to be a subdirectly irreducible (or cococyclic) submodule of $M$ if the quotient module $M/N$ is subdirectly irreducible. These concepts can be naturally relativized with respect to a hereditary torsion theory $t$ on Mod-$R$, or more generally, can be defined for arbitrary posets with 0 and 1.

As a module is the sum of all its cyclic modules, dually, for any module $M$, the intersection of all subdirectly submodules of $M$ is zero; one also says that the lattice of all submodules of $M$ is rich in subdirectly irreducibles. The relative version of this property does not hold in general.

In this talk we characterize right modules $M$ having relative $t$-dual Krull dimension less than or equal to a given ordinal. This requires that $M$ is rich in $t$-subdirectly irreducibles. Therefore, we first look for sufficient conditions on the torsion theory $t$ to insure that, for any module $M$, the lattice of all $t$-closed submodules of $M$ is rich in subdirectly irreducibles. In particular we give a relative version to an extension in the more general setting of dual Krull dimension of a nice recent result due to Carl Faith saying that a right $R$-module $M$ is Noetherian if and only if $M$ is QFD and satisfies the ACC on subdirectly irreducible submodules.

The results to be presented, have been obtained jointly with Mihai Iosif (Bucharest University, Romania) and Mark L. Teply (University of Wisconsin-Milwaukee, USA).

RIGHT-LEFT SYMMETRY OF RIGHT NONSINGULAR

RIGHT MAX-MIN $CS$ PRIME RINGS

HUSAIN S. AL-HAZMI, Ohio University

Abstract

A ring $R$ is called right (left) max-min $CS$-ring if every maximal closed right (left) ideal with nonzero left (right) annihilator and every minimal closed right (left) ideal of $R$ is a direct summand of $R$. In this paper we show that if $R$ is a prime ring which is not a domain, then $R$ is right nonsingular, right max-min $CS$-ring with uniform right ideal if and only if $R$ is left nonsingular, left max-min $CS$-ring with uniform left ideal. The above result gives, in particular, Huynh-Jain-López Theorem [Proc. Amer. Math. Soc. 128, 3153-3157 (2000)] (joint work with Adel N. Alahmadi and S. K. Jain).

ON GABRIEL'S CONDITION $H$

JOHN BEACH, Northern Illinois University

Abstract

(Preliminary report)

We investigate modules which satisfy an analog in $\sigma[M]$ of the "condition H" introduced in Gabriel's thesis.

SOME COMMUTATIVITY-OR-FINITENESS

CONDITIONS FOR RINGS

HOWARD E. BELL, Brock University

ABRAHAM A. KLEIN, Tel Aviv University

Abstract

Fifteen years ago, Bell and Guerriero conjectured that a ring having only a finite number of noncentral subrings must be either finite or commutative. We have recently found a proof of this result, which involves some unexpected notions. We give an indication of the proof, and we discuss some related results.

RING AND MODULE HULLS

GARY F. BIRKENMEIER, University of Louisiana - Lafayette

Abstract

(This talk is based on joint work with Jae Keol Park and S. Tariq Rizvi).

In this talk we discuss various concepts for the hull of a ring or module, where the hull is from a certain class of rings or modules, respectively. Existence and/or uniqueness results on hulls from various classes (including extending, FI-extending, continuous, Baer, etc.) are presented. Also the transfer of information between an object and its hull is considered. Examples are provided to illustrate the theory.

RINGS WHOSE CYCLIC MODULE HAVE

$\Sigma$-CS INJECTIVE HULL

S. CHAIRAT, Mahidol University, Bangkok, Thailand

Abstract

We consider conditions that make the rings in the title Noetherian.

TYPE SUBMODULES AND MODULE

DECOMPOSITION

JOHN DAUNS, Tulane University

YIQIANG ZHOU, Memorial University of Newfoundland, St.John's, Canada

Abstract

A type decomposition of a module $M$ over a ring $R$ is a direct sum decomposition for which any two distinct summands have no nonzero isomorphic submodules. In this paper, we investigate when a module possesses certain kinds of type decompositions and when such decompositions are unique.

A class of right $R$-modules $\mathcal K$ of modules is a type (or natural class) if it is closed under isomorphic copies, submodules, arbitrary direct sums and injective hulls. A submodule $N$ of $M$ is a type submodule if, for some type $\mathcal K$, $N$ is a submodule of $M$ which is maximal with respect to $N\in \mathcal K$. In this case, we also say $N$ is a type submodule of type $\mathcal K$. Just as every submodule $N<M$ has a complement closure in $M$, so also it has a type closure $N\subseteq N^{tc}\leq M$. Just as in the complement case, $N^{tc}$ need not be unique. A module $M$ satisfies UTC (unique type closure) if every submodule $N<M$ has a unique type closure $N\subseteq N^{tc}\leq M$, or equivalently if for every type $\mathcal K$, $M$ has a unique type submodule of type $\mathcal K$. Several equivalent characterizations of UTC modules are obtained. Fully invariant type submodules and fully invariant type direct summands are explored.

AN APPLICATION OF GALOIS RINGS

IN ALGEBRAIC CODING THEORY

HAI QUANG DINH, North Dakota State University

Abstract

Galois rings have been used widely in Algebraic Coding Theory. In this talk, we will discuss negacyclic codes of length $2^s$ over Galois rings. It will be shown that $\dfrac{GR(2^a, m)} {\langle x^{2^s} + 1 \rangle}$ is indeed a chain ring, and hence obtaining the complete structure of negacyclic codes of length $2^s$ over $GR(2^a, m)$, as well as that of their duals. We also address the minimum Hamming weights of such negacyclic codes. Open directions for further investigations will also be discussed.

MODULES WITH CHAIN CONDITIONS

ON ENDOSUBMODULES

NGUYEN VIET DUNG, Ohio University - Zanesville

Abstract

A right module $M$ over a ring $R$ (with identity) is called endofinite if it has finite length as a left module over its endomorphism ring. It is well-known that endofinite modules have many attractive properties: Every endofinite module $M$ is $\Sigma$-pure-injective, with nilpotent Jacobson radical of the endomorphism ring End$_RM$, and admits an indecomposable decomposition $M = \oplus _{i\in I}M_i$, where the number of isomorphism classes of the $M_i$ is finite. In this talk, we will discuss modules satisfying some weaker forms of chain conditions on endosubmodules, and show that a number of the above properties of endofinite modules can be extended to our situation. Among the applications, we show that any pure-injective pure-projective module over an Artin algebra is endo-Artinian. (A part of the results of this talk is based on our joint work with Sergio R. López-Permouth).

PURE INJECTIVITY FOR LOCALLY

FINITELY PRESENTED CATEGORIES II

PEDRO A. GUIL ASENSIO, University of Murcia, Murcia, Spain

Abstract

It is proved that a ring $R$ is right $\Sigma$-pure injective in the category of flat modules if and only if it is right perfect. Several other conditions are shown to be equivalent. For example, that a countable direct sum of copies of its pure-injective envelope in Flat-$R$ is pure-injective in that category. (Joint work with Ivo Herzog).

HYPER-RADICAL AND THE LEVITSKI-HOPKINS

THEOREM FOR MODULAR LATTICES

FERNANDO GUZMAN, Binghamton University

Abstract

Many arguments in the Theory of Rings and Module are, on close inspection, purely Lattice theoretic arguments. Grigore Calugareanu has a long repertoire of such results in his book "Lattice Concepts on Module Theory". The Levitski-Hopkins Theorem is interesting from this point of view, because only part of it lends to an obvious lattice theory approach. The rest of it requires a new construction, the Hyper-radical, which will be presented in this talk.

PURE INJECTIVITY FOR LOCALLY

FINITELY PRESENTED CATEGORIES I

IVO HERZOG, The Ohio State University-Lima

Abstract

It is shown that a ring is right cotorsion if and only if it is the endomorphism ring of a pure-injective object in a locally finitely presented additive category. It is also shown that these rings are (von Neumann) regular and self injective module their Jacobson radical and idempotents lift. (Joint work with Pedro A. Guil Asensio).

GROUP RINGS AND GENERALIZATIONS

OF INJECTIVITY

PRAMOD KANWAR, Ohio University-Zanesville

Abstract

A ring $R$ is said to be right CS if every right ideal is essential in a summand of $R$. A right CS ring is said to be right continuous if any right ideal of $R$ which is isomorphic to a summand of $R$ is itself a summand of $R$. Right CS rings and right continuous rings are generalizations of right selfinjective rings. It is known that the group algebra $KG$ of a group $G$ over a field $K$ is selfinjective if and only if $G$ is a finite group. The group algebra $KG$ is principally injective if and only if $G$ is a locally finite group. We discuss group algebras that are continuous (or CS) and give the corresponding conditions on the group $G$. Among others, it is shown that if the group algebra $KG$ of a group $G$ over a field $K$ is right continuous then $G$ is a locally finite group. In particular, a right continuous group algebra is principally injective.

CERTAIN RINGS WHOSE SIMPLE

SINGULAR MODULES ARE $GP$-INJECTIVE

JIN YONG KIM, Kyung Hee University, Suwon, South Korea

Abstract

We investigate in this paper von Neumann regularity of certain rings whose simple singular left $R$-modules are $GP$-injective. As a nontrivial generalization of a reflexive ring, idempotent reflexive ring is considered. It is proved that if $R$ is an idempotent reflexive ring whose simple singular left $R$-modules are $GP$-injective, then for any nonzero element $a$ in $R$, there exists a positive integer $n=n(a)$ such that $a^n\neq 0$ and $RaR+l(a^n)=R$. Several known results are extended and unified.

TORSION-FREE RANKS OF INDECOMPOSABLE

MODULES

LEE KLINGLER, Florida Atlantic University

Abstract

In joint work with L. Levy, we showed that a commutative, Noetherian ring $R$ has tame representation type if and only if $R$ is a Klein ring or a homomorphic image of a Dedekind-like ring.

Klein rings have finite length, while Dedekind-like rings are reduced and one-dimensional and have the property that all of their finitely generated indecomposable modules have torsion-free rank at most two. Following this observation, R. Wiegand conjectured that, for a commutative, Noetherian, reduced, one-dimensional ring $R$, there is a bound on the torsion-free rank of finitely generated indecomposable $R$-modules if and only if $R$ has tame representation type (if and only if $R$ is a Dedekind-like ring).

In joint work with W. Hassler, R. Karr, and R. Wiegand, we prove this conjecture and more.

ON RIGHT ANNELIDAN RINGS

GREG MARKS, St. Louis University

Abstract

We say a ring is "right annelidan" if the right annihilator of any element of the ring is a right waist. This class of rings encompasses domains as well as right uniserial rings and certain other local rings. We obtain results on the structure of right annelidan rings, their ideals, and important radicals. We show the Köthe Conjecture holds for this class of rings, and we study the relationship between this and various other major classes of rings (joint work with Ryszard Mazurek).

RING STRUCTURES OF INJECTIVE

HULLS

JAE KEOL PARK, Busan National University, Pusan, South Korea

Abstract

We discuss a right injective hull $E$ of a ring $R$ such that $E$ has exactly four distinct ring structures whose multiplications extend the module multiplication of $E$ over $R$ and all these ring structures are isomorphic. This gives a negative answer to Osofsky’s question on the uniqueness of the compatible ring multiplication for the injective hull. Various types of ring hulls of $R$ and all intermediate ring structures between $R$ and $E$ are discussed. (jointly done with Gary F. Birkenmeier and S. Tariq Rizvi).

WHEN EVERY PROJECTIVE MODULE

IS A DIRECT SUM OF FINITELY

GENERATED MODULES

GENA PUNINSKI, The Ohio State University-Lima

Abstract

We discuss old and prove new results on rings over which every projective right module is a direct sum of finitely generated modules.

By Albrecht and Bass this is true for left or right semihereditary rings. It follows from a generalization of Krull-Azumaya-Schmidt theorem that semiperfect rings enjoy this property. By Harada the same is true for every commutative weakly noetherian ring.

We give a precise criterion (in terms of matrices over a ring) when every projective right module is a direct sum of finitely generated modules.

Outside the above mentioned classes of rings, the answer to this question is widely unknown. For instance, it is not known if every projective module over a commutative domain is a direct sum of finitely generated modules.

In particular, we give different examples of noetherian rings where the answer to the above question is 1) negative; 2) positive; and 3) unknown.

For instance, as a byproduct of our results we give a complete classification of non-finitely generated projective modules over all primitive factors of $Usl_2(k)$ ($k$ is an algebraically closed field of characteristic zero) and over the ring of differential operators on $\mathbb{P}^{\, n}$, the projective space.

VAUGHT'S CONJECTURE AND MODULES

WITH FEW TYPES

VERA PUNINSKAYA, University of Camerino, Camerino, Italy

Abstract

The following statement is known as Vaught's conjecture:

Conjecture. Let $T$ be a countable complete first-order theory. If $T$ has uncountably many non-isomorphic countable models, then $T$ has $2^{\omega}$ non-isomorphic countable models.

Garavaglia proved Vaught's conjecture for $\omega$-stable theories of modules in 1980. In 1984 Shelah extended this result to arbitrary $\omega$-stable theories.

The main objective of this talk is to show that Vaught's conjecture is true for any complete theory $T$ of modules over 1) countable serial rings; 2) countable commutative Prüfer rings; 3) countable hereditary noetherian prime rings and 4) tame hereditary finite-dimensional algebras over a countable infinite field. In all these cases we prove that, if $T$ has few types, then it is $\omega$-stable, making it possible to apply Garavaglia's result.

We give a complete classification of modules with few models over countable serial rings, and of modules with few types over countable commutative Prüfer rings. A similar description holds true for modules over a string algebra $K\Lambda_2$ over a finite or countable field $K$.

We also discuss some partial results on Vaught's conjecture for modules over group rings and for modules over pullback rings.

TYPE THEORY FOR NONSINGULAR CS MODULES.

PRELIMINARY REPORT

COSMIN ROMAN, The Ohio State University

Abstract

Recently we defined the concept of right Baer modules over a ring $R$. A module $M$ is called Baer if the right annihilator of any set of endomorphisms of $M$, in $M$ is a direct summand of $M$. This concept generalizes the concept of Baer rings. Kaplansky developed a theory of types for Baer rings, and Goodearl and Boyle extended it to nonsingular injective modules. We extend this type theory to Baer modules and, in particular, since nonsingular CS modules are Baer, to nonsingular CS modules (joint work with S.T. Rizvi).

ON HARADA RINGS AND

SERIAL RINGS

NGUYEN VAN SANH, Mahidol University, Bangkok, Thailand

Abstract

A ring $R$ is called a right Harada ring if it is right Artinian and every non-small right $R$-module contains a non-zero injective submodule. The first result in our report is the following Theorem: Let $R$ be a right perfect ring. Then $R$ is a right Harada ring if and only if every cyclic module is a direct sum of an injective module and a small module; if and only if every localmodule is either injective or small. We also prove that a ring $R$ is QF if and only if every cyclic module is a direct sum of a projective injective module and a small module; if and only if every local module is either projective injective or small. Finally, a right QF-3 right perfect ring $R$ is serial if and only if every right ideal is a direct sum of a projective module and a singular module.

SUBMODULES OF MODULES AND AN

ENDOFUNCTOR OF PERIOD 6

MARKUS SCHMIDMEIER, Florida Atlantic University

Abstract

For $R$ an algebra, consider the category $S(R)$ of all pairs $(B,A)$ where $B$ is a finite length $R$ -module and $A$ a submodule of $B$. Then $S(R)$ is a full and exact subcategory of a module category, so every object in $S(R)$ is a direct sum of indecomposables. It turns out that $S(R)$ has Auslander-Reiten sequences, so we computed a formula for the Auslander-Reiten translation, which acts on the set of indecomposables in $S(R)$. Surprisingly, for $R$ a uniserial ring, this translation has period six. This is a report on joint work with Claus Michael Ringel.

SOME NOTES ON THE STRUCTURE

OF A GALOIS ALGEBRA

GEORGE SZETO, Bradley University

Abstract

(Joint work with Lianyond Xue)

Let $B$ be a Galois algebra over a commutative ring $R$ with Galois group $G$. Then $B$ is a direct sum of Galois algebras such that each direct summand is a composition of a central Galois algebra and a commutative Galois algebra. A sufficient condition is also given under which $B$ is commutative.