Aspects of Infinite Groups: A Festschrift in Honor of Anthony Gaglione (Algebra and Discrete Mathematics)

253 Pages·2008·10.29 MB·English

by  Benjamin Fine

Algebra and Discrete Mathematics
A Festschrift in Honor of Anthony Gaglione
Algebra and Discrete Mathematics ISSN: 1793-5873
Managing Editor: RUdiger Gobel (University Duisburg-Essen, Germany)
Editorial Board: Elisabeth Bouscaren, Manfred Droste, Katsuya Eda,
Emmanuel Dror Farjoun, Angus MacIntyre, H.Dugald Macpherson,
Jose Antonio de la Pefia, Luigi Salce, Mark Sapir, Lutz Strlingmann,
Simon Thomas
The series ADM focuses on recent developments in all branches of algebra and
topics closely connected. In particular, it emphasizes combinatorics, set theoretical
methods, model theory and interplay between various fields, and their influence on
algebra and more general discrete structures. The publications ofthis series are of
special interest to researchers, post-doctorals and graduate students. It is the
intention of the editors to support fascinating new activities of research and to
spread the new developments to the entire mathematical community.
Vol. I: Aspects of Infinite Groups: A Festschrift in Honor of Anthony Gaglione
eds. Benjamin Fine, Gerhard Rosenberger & Dennis Spellman
Algebra and Discrete Mathematics
ASPECTS
A
I"
INIT~ G~O ~
A Festschrift in Honor of Anthony Gaglione
editors
Benjamin Fine
Fairfield University, USA
Gerhard Rosenberger
Universitat Dortmund, Germany
Dennis Spellman
Temple University, USA
World Scientific
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Algebra and Discrete Mathematics - Vol. 1
ASPECTS OF INFINITE GROUPS
A Festschrift in Honor of Anthony Gaglione
Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd.
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ISBN-13 978-981-279-340-9
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ASPECTS OF INFINITE GROUPS
Editors:
Benjamin Fine, Gerhard Rosenberger, Dennis Spellman
PREFACE
This volume consists of contributions by participants and speakers at the
conference entitled Aspects of Infinite Groups held at Fairfield University
in March 2007 in honor of Prof. Anthony Gagliones sixtieth birthday.
Prof. Gaglione of the United States Naval Academy at Annapolis has
made important and varied contributions to several different areas of Infi
nite Group Theory and Combinatorial Group Theory. Along with Herman
Waldinger he developed and studied many important topics within the
commutator calculus. Along with Dennis Spellman he proved a magnifi
cent result tying together the concepts of residual freeness and universal
freeness. This result was also done independently by Vladmir Remeslen
nikov and can be considered as one of the seminal steps in the final proof
of the celebrated Tarski problems. The final proof of the Tarski problems
was accomplished by O.Kharlampovich and A.Myasnikov and indepen
dently by Z.Sela. Finally Prof. Gaglione, along with G.Baumslag, B.Fine,
A.Myasnikov, V.Remeslennikov and D.Spellman completely developed the
theory of discriminating and sqaurelike groups. This important class of
groups was introduced by G.Baumslag, A.Myasnikov and V.Remeslennikov
as an outgrowth of the development of algebraic geometry over groups.
The papers in this volume provide an interesting mix of results on
modern infinite discrete group theory ranging from classical combinato
rial group theory to algebraic geometry over groups to noncommutative
algebraic cryptography.
The main speakers were primarily people who worked closely with Prof.
Gaglione. They were
Prof. Michael Anshel
City University of New York, New York City, NY
Prof. Gilbert Baumslag
City University of New York, New York City, NY
Prof. Benjamin Fine
v
vi
Fairfield University, Fairfield, CT
Prof. Alexei Myasnikov
McGill University, Montreal, Canada
Prof. Gerhard Rosenberger
TU Dortmund, Dortmund, Germany
Prof. Dennis Spellman
Temple University, Philadelphia, Pennsylvania
Prof. Gaglione received his Ph.D. in Mathematics from Polytechnic Uni
versity in Brooklyn, New York in 1972. He has been a professor at the
United States Naval Academy since 1977.
He is the author or coauthor of more than 70 journal articles represent
ing the areas of combinatorial group theory described earlier. He is also the
author or coauthor of four books.
Benjamin Fine
Gerhard Rosenberger
Dennis Spellman
CONTENTS
Preface v
Actions, Commutator Identities, and the Algebraic Eraser™ 1
I. Anshel, M. Anshel and D. Goldfeld
Virtually Free-By-Cyclic One-Relator Groups: I 9
G. Baumslag and D. Troeger
Some Cryptoprimitives for Noncommutative Algebraic
Cryptography 26
G. Baumslag, Y. Bryukhov, B. Fine and G. Rosenberger
On the Derived Subgroups of the Free Nilpotent Groups of
Finite Rank 45
R. D. Blyth, P. Moravec and R. F. Morse
A Recurrence Relation for the Number of Free Subgroups in
Free Products of Cyclic Groups 54
T. Camps, M. Darfer and G. Rosenberger
The Baumslag-Solitar Groups: A Solution for the Isomorphism
Problem 75
A. E. Clement
Unification Theorems in Algebraic Geometry 80
E. Daniyarova, A. Myasnikov and V. Remeslennikov
Reflections on Commutative Transitivity 112
B. Fine and G. Rosenberger
Groups Universally Equivalent to Free Burnside Groups of
Prime Exponent and a Question of Philip Hall 131
A. Gaglione, S. Lipschutz and D. Spellman
Changing Generators in Nonfree Groups 149
R. Goldstein
vii
viii
Matrix Completions Over Principal Ideal Rings 151
W. H. Gustafson, D. W. Robinson, R. B. Richter
and W. P. Wardlaw
A Primer on Computational Group Homology and Cohomology 159
D. Joyner
Doubles of Residually Solvable Groups 192
D. K ahrobaei
An Application of a Group of Ol'shanskii to a Question of
Fine et. aZ. 201
S. Lipschutz and D. Spellman
Quotient Isomorphism Invariants of a Finitely Generated
Coxeter Group 212
M. Mihalik, J. Ratcliffe and S. Tschantz
Localization and I A-automorphisms of Finitely Generated,
Metabelian, and Torsion-Free Nilpotent Groups 228
M. Zyman
ACTIONS, COMMUTATOR IDENTITIES, AND THE
ALGEBRAIC ERASER™
Iris Anshel
31 Peter Lynas Ct. Tenafly, NJ 07670, USA
Michael Anshel
City College of New York, New York, NY 10031, USA
Dorian Goldfeld'
Columbia University, Department of Mathematics, New York, NY 10027
Dedicated to Tony Gaglione on his 60th birthday.
Abstract: An algebraic structure arising in the formulation of a lightweight
key agreement protocol yieldsa new concept of commutator whose identities
are quite traditional
1. Introduction
In [AAGL ] the authors introduced a key agreement protocol for public
key cryptography suitable for implementation on lightweight platforms,that
is, those subject to severe cost and resource constraints. Careful examina
tion reveals a hidden notion of commutator possessing identities analagous
to those formulated by P. Hall, W. Magnus, E. Witt. The question as to
whether or not these eminent mathematicians of the twentieth century de
veloped the theory of commutators for cryptographic purposes was raised in
[AG]. Our method is to formulate the necessary mathematical primitives
as monoid (group) actions and then identify the objects of interest. We
conclude by inviting the reader to explore a certain action in the context of
'The authors would like to thank SecureRF for its support of this research