Computational Group Theory and the Theory of Groups II: Computational Group Theory and Cohomology, August 4-8, 2008, Harlaxton College, Grantham, ... Group Theor

210 Pages·2010·3.508 MB·English

by  Luise-charlotte Kappe, Arturo Magidin, Robert Fitzgerald Morse (ed.)

C
ONTEMPORARY
M
ATHEMATICS
511
Computational Group Theory
and the Theory of Groups, II
Computational Group Theory and Cohomology
August 4–8, 2008
Harlaxton College, Grantham, United Kingdom
AMS Special Session
Computational Group Theory
October 17–19, 2008
Western Michigan University
Kalamazoo, MI
Luise-Charlotte Kappe
Arturo Magidin
Robert Fitzgerald Morse
Editors
American Mathematical Society
Computational Group Theory
and the Theory of Groups, II
This page intentionally left blank
C
ONTEMPORARY
M
ATHEMATICS
511
Computational Group Theory
and the Theory of Groups, II
Computational Group Theory and Cohomology
August 4–8, 2008
Harlaxton College, Grantham, United Kingdom
AMS Special Session
Computational Group Theory
October 17–19, 2008
Western Michigan University
Kalamazoo, MI
Luise-Charlotte Kappe
Arturo Magidin
Robert Fitzgerald Morse
Editors
American Mathematical Society
Providence, Rhode Island
Editorial Board
Dennis DeTurck, managing editor
George Andrews Abel Klein Martin J. Strauss
2000 Mathematics Subject Classification. Primary 20–06, 20B40, 20B35,17B55, 18G10,
20F12, 20F18, 20H15, 20J99, 20P05.
The photograph of Harlaxton College appears courtesy of the Universityof Evansville
Library of Congress Cataloging-in-Publication Data
HarlaxtonConferenceonComputationalGroupTheoryandCohomology(2008: HarlaxtonCol-
lege)
Computationalgrouptheoryandthetheoryofgroups,II:HarlaxtonConferenceonComputa-
tionalGroupTheory andCohomology,HarlaxtonCollege,the BritishCampus ofthe University
ofEvansville,Grantham,UnitedKingdom,August4–8,2008: AMSSpecialSessiononComputa-
tionalGroupTheory,WesternMichiganUniversity,Kalamazoo,Michigan,October17–19,2008/
Luise-CharlotteKappe,ArturoMagidin,RobertFitzgeraldMorse,editors.
p.cm. —(Contemporarymathematics;v.511)
Includesbibliographicalreferences.
ISBN978-0-8218-4805-0(alk.paper)
1.Grouptheory—Congresses. 2.Computationalcomplexity—Congresses. I.Kappe,Luise-
Charlotte. II. Magidin, Arturo. III. Morse, Robert Fitzgerald. IV. AMS Special Session on
ComputationalGroupTheory(2008: WesternMichiganUniversity) V.Title.
QA174.H37 2010
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Contents
Preface vii
The probabilistic zeta function
Bret Benesh 1
Periodicities for graphs of p-groups beyond coclass
Bettina Eick and Tobias Rossman 11
Computing covers of Lie algebras
Graham Ellis, Hamid Mohammadzadeh, and Hamid Tavallaee 25
Enumerating subgroups of the symmetric group
Derek F. Holt 33
Weight five basic commutators as relators
David A. Jackson, Anthony M. Gaglione,
and Dennis Spellman 39
Basic commutators as relations: A computational perspective
Primoˇz Moravec and Robert Fitzgerald Morse 83
Groups of minimal order which are not n-power closed
Luise-Charlotte Kappe and Gabriela Mendoza 93
On the covering number of small alternating groups
Luise-Charlotte Kappe and Joanne L. Redden 109
Certain homological functors of 2-generator p-groups of class 2
Arturo Magidin and Robert Fitzgerald Morse 127
Geometric algorithms for resolutions for Bieberbach groups
Marc Ro¨der 167
Nonabelian tensor product of soluble minimax groups
Francesco Russo 179
Finite groups have short rewriting systems
Jack Schmidt 185
v
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Preface
This is a follow-up to the Contemporary Mathematics volume that we edited
on the same topic and that was published by the American Mathematical Society
in 2008. This volume consists of contributions by researchers who were invited to
the Harlaxton Conference on Computational Group Theory and Cohomology, held
at Harlaxton College, The British Campus of the University of Evansville, August
4-8,2008;andtotheAMSSpecialSessiononComputationalGroupTheoryheldat
WesternMichiganUniversity, October17-19, 2008. The HarlaxtonConferencewas
supported financially by the De Brun Centre for Computational Algebra, National
UniversityofIreland,Galway,andtheUniversityofEvansville’sInstituteforGlobal
Enterprise in Indiana.
BoththeConferenceandtheSpecialSessionfocusedonexamplesofusingCGT
to solve problems that arise from many areas of group theory; in this volume we
find applications to the enumeration of subgroups of the symmetric group, cover-
ing groups by subgroups, the ongoing co-class project for classification of p-groups,
constructionorcomputationofhomologicalandcohomologicalinvariantsofgroups,
probabilistic group theory, and the study of free groups, among others. Compu-
tational Group Theory plays many roles in these investigations, from exploration
thatsuggestsconjecturesor proofs, throughperforming keycomputationsrequired
to establish theorems.
Once again, we present these examples in the hope that they will encourage
researchers and graduate students to think about ways in which they can incorpo-
rate CGT in their own research by seeing many different applications of CGT to
traditional problems in Group Theory.
The Harlaxton Conference was organized by Bettina Eick, Graham Ellis, and
Robert F. Morse; we thank them very much for all their work, and we also thank
the de Brun Centre and the University of Evansville for their financial support.
The second editor was supported in part by a grant from the Louisiana Board
of Regents. The first and third editors thank Arturo for his work managing and
editing the submissions. The three of us are grateful to all the participants in the
conferences, and to all authors who submitted contributions to this volume. We
are also very thankful indeed to the referees who did such an excellent and timely
job for both this and the previous volume. Finally, we are also very grateful to the
American Mathematical Society for their help in the publication of this volume,
particularly to Christine M. Thivierge for her help and patience.
Luise-Charlotte Kappe
Arturo Magidin
Robert Fitzgerald Morse
vii

CCoonntteemmppoorraarryyMMaatthheemmaattiiccss
Volume511,2010
The Probabilistic Zeta Function
Bret Benesh
Abstract. ThispaperisasummaryofresultsonthePG(s)function,whichis
thereciprocaloftheprobabilisticzetafunctionforfinitegroups. Thisfunction
givestheprobabilitythatsrandomlychosenelementsgenerateagroupG,and
informationaboutthestructureofthegroupGisalsoembeddedinit.
1. Introduction and History
Probabilisticgrouptheoryhasbeenagrowingfieldofmathematicsforthepast
coupleofdecades. Whileotherpapershaveconsideredthisfieldingreatergenerality
(see[Di,Shal1,Shal2]), wewillbefocusingontheso-called P (s)function, which
G
is the function that gives the probability that s randomly chosen elements (with
replacement) of a finite group G generate G.
The study of the P (s) function began in 1936, when Philip Hall [H] created
G
the Eulerian functionφ (s),defined tobe the number of s-tuples (g ,...,g )∈Gs
G 1 s
such that (cid:2)g ,...,g (cid:3)=G. Hall showed that
1 s
(cid:1)
φ (s)= µ (H)|H|s,
G G
H≤G
whereµ (H)istheMo¨biusfunctionofthesubgrouplatticeofG,definedinductively
G
as µ (G)=1 and
G (cid:1)
µ (K)=0
G
H≤K≤G
if H <G.
AfterHall,G.E.Wall[W] usedavariationoftheEulerianfunction(theEuler-
ian polynomial) to prove the following theorem.
Theorem 1.1 (Wall’sTheorem). If G is a finite solvable group, then the num-
ber of maximal subgroups in G is less than |G|.
2000MathematicsSubjectClassification. 20E34,20F05,20P05,11M41.
Key words and phrases. group theory, zeta functions, Dirichlet series, subgroup lattices,
Moebiusfunctions.
TheauthorwouldliketothankNigelBoston,ErikaDamian,andtherefereefortheirthought-
fulcommentsonthepaper.
(cid:1)c2010 Americ(cid:1)acn00M00at(hceompyartiigcahltShoolcdieetry)
11