2018 zassenhaus groups and friends conference
TRANSCRIPT
2018 Zassenhaus Groups and Friends ConferenceUniversity of South Florida - Tampa - FLApril 6-8, 2018
The conference continues the series of Ohio State-Denison Mathematics Conferences first organized in the 1960’s by Hans Zassenhaus.
Organizers:E. Ahmed (USF)
F. Guzman (Binghamton U)A. Magidin (U Louisianna Lafayette)
D. Savchuk (USF)E. Wilcox (SUNY Oswego)
Web:http://www.math.usf.edu/ZGFC/
Sponsored by:USF Internal Awards Program
Department of Mathematics and Statistics (USF)
This year the conference is dedicated to:
and memory of Jim Biedleman
the 75th anniversary of Ben Brewster
Representation Theory
Universal Algebras
Graph Theory
and other related areas
Group TheoryLoop Theory
2018 Zassenhaus Group Theory and Friends ConferenceUniversity of South Florida, Tampa, FL
April 6–8, 2018
PROGRAM &ABSTRACTS
ORGANIZERS
Elsayed Ahmed, University of South Florida, Tampa, FloridaFernando Guzman, Binghamton University, Binghamton, New YorkArturo Magidin, University of Louisiana at Lafayette, Lafayette, LouisianaDmytro Savchuk, University of South Florida, Tampa, FloridaElizabeth Wilcox, State University of New York at Oswego, Oswego, New York
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Fri Apr 6 – Sun Apr 8, 2018 (Eastern Time)Zassenhaus Conference
Lunch12pm - 1 :30pm
Coffee Break2:30pm - 3pm
Coffee Break10am - 10 :30am
Coffee Break3pm - 3 :30pm
Coffee Break10am - 10 :30am
Coffee Break10am - 10 :30am
Grigorchuk1:30pm - 1 :50pm
Kappe10:30am - 10 :50am
Ferrar i5pm - 5 :20pm
Sunic4:30pm - 4 :50pm
Samarakoon2pm - 2 :20pm
Shahrtash3:30pm - 3 :50pm
Margolis9am - 9 :20am
Longobardi11am - 11 :20am
Kravchenko1:30pm - 1 :50pm
H o u2pm - 2 :20pm
Burket t2pm - 2 :20pm
Pantangi1:30pm - 1 :50pm
Lynd3pm - 3 :20pm
Seki2:30pm - 2 :50pm
Kogan4pm - 4 :20pm
McColm3:30pm - 3 :50pm
McCulloch9:30am - 9 :50am
Crockett11:30am - 11 :50am
Ealy Jr.9am - 9 :20am
Curtin9:30am - 9 :50am
Dranishnikov10:30am - 10 :50am
Zappala11am - 11 :20am
Robinson11:30am - 11 :50am
Nekrashevych2:30pm - 2 :50pm
Benesh9am - 9 :20am
Ahmed9:30am - 9 :50am
Juschenko10:30am - 10 :50am
Karatas11am - 11 :20am
Turull11:30am - 11 :50am
Breakfast (EDU 314)8am - 8 :30am
Breakfast (EDU 314)8am - 8 :30am
Elhamdadi (EDU 347)8:30am - 8 :50am
Atanasov (EDU 347)8:30am - 8 :50am
Banquet (TECO hall, EDU building)6:30pm - 8 :30pm
Registration, Breakfast (MSC 2709)8:20am - 9am
Lunch (EDU 314)12pm - 1 :30pm
Lunch (EDU 314)12pm - 1 :30pm
Fri 4 /6 Sat 4 /7 Sun 4/88am
9am
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Friday, April 6, 2018
Location: Marshall Student Center (MSC) - 2709
8:20 - 9:00 AM Registration, Continental Breakfast, and WelcomeMSC 2709
9:00 - 9:20 AM Leo Margolis (Free University of Brussels, Belgium)A counterexample to the first Zassenhaus conjecture
9:30 - 9:50 AM Ryan McCulloch (University of Bridgeport)A result on the Chermak-Delgado lattice of a finite group
10:00 - 10:30 AM Coffee Break
10:30 - 10:50 AM Luise-Charlotte Kappe (Binghamton University)On integers that are covering numbers of groups
11:00 - 11:20 AM Patrizia Longobardi (Universita di Salerno, Italy)Two new criteria for solvability of finite groups
11:30 - 11:50 AM Eran Crockett (Binghamton University)A characterization of solvable A-groups and its generalization
to universal algebra
12:00 - 1:30 PM LUNCH
MSC 2709
1:30 -1:50 PM Venkata Raghu Tej Pantangi (University of Florida)Smith and critical groups of polar graphs
2:00 - 2:20 PM Xiang-dong Hou (University of South Florida)The Mobius function of the affine linear group AGL(1,Fq)
2:30 - 3:00 PM Coffee Break
3:00 - 3:20 PM Justin Lynd (University of Louisiana at Lafayette)The Benson-Solomon fusion systems
3:30 - 3:50 PM Hossein Shahrtash (University of Florida)Rational class sizes and their implications about
the structure of a finite group
Saturday, April 7, 2018
Location: EDU 347
8:00 - 8:30 AM Continental Breakfast8:30 - 8:50 AM Risto Atanasov (Western Carolina University)
Bounds of nilpotency class of finite p-groups9:00 - 9:20 AM Clifton Ealy, Jr. (Western Michigan University)
Matrix loops over proper Kalscheuer near-fields
9:30 - 9:50 AM Brian Curtin (University of South Florida)An Euler totient sum inequality
10:00 - 10:30 AM Coffee Break
EDU 314
10:30 - 10:50 AM Alexander Dranishnikov (University of Florida)On the Burghelea conjecture
11:00 - 11:20 AM Emanuele Zappala (University of South Florida)Distributive groupoids and their cohomologies
11:30 - 11:50 AM Derek Robinson (University of Illinois at Urbana-Champaign)On groups with countably many maximal subgroups
12:00 - 1:30 PM LUNCH
EDU 314
1:30 - 1:50 PM Rostislav Grigorchuk (Texas A&M University)Invariant random subgroups and factor representations of
branch groups
2:00 - 2:20 PM Supun Samarakoon (Texas A&M University)On growth of generalized Grigorchuk’s overgroups
2:30 - 2:50 PM Volodymyr Nekrashevych (Texas A&M University)Simple groups of intermediate growth
3:00 - 3:30 PM Coffee Break
EDU 314
3:30 - 3:50 PM Gregory McColm (University of South Florida)Enumerating periodic graphs
4:00 - 4:20 PM Roman Kogan (Industry / Former Texas A&M University)Finite-state automata and measures
4:30 - 4:50 PM Zoran Sunic (Hofstra University)Rewriting in Thompson’s group F
5:00 - 5:20 PM Margherita Maria Ferrari (University of South Florida)Mathematical models for describing molecular self-assembly
6:30 - 8:30 PM Banquet
TECO Hall in EDU building
Sunday, April 8, 2018
Location: EDU 347
8:00 - 8:30 AM Continental Breakfast8:30 - 8:50 AM Mohamed Elhamdadi (University of South Florida)
Quandles and groups
9:00 - 9:20 AM Bret Benesh (College of St. Benedict and St. John’s University)The game GENERATE on finite nilpotent groups
9:30 - 9:50 AM Elsayed Ahmed (University of South Florida)Endomorphisms of regular rooted trees induced by
the action of polynomials on the ring Zd of d-adic integers
10:00 - 10:30 AM Coffee Break
EDU 314
10:30 - 10:50 AM Kate Juschenko (Northwestern University)On Liouville property of action of discrete groups
11:00 - 11:20 AM Zekeriya Karatas (University of Cincinnati Blue Ash College)Groups whose non-permutable subgroups are soluble minimax
11:30 - 11:50 AM Alexandre Turull (University of Florida)The invariant of a character
12:00 - 1:30 PM LUNCH
EDU 314
1:30 - 1:50 PM Rostyslav Kravchenko (Northwestern University)On the rigidity of rank gradient in a group of intermediate
growth
2:00 - 2:20 PM Shawn Burkett (University of Colorado Boulder)A generalized version of nilpotence arising from
supercharacter theory
2:30 - 2:50 PM Shinnosuke Seki (University of Electro-Communications, Tokyo)Proving Turing universality of cotranscriptional folding
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Local Information
Places to Eat:
• Saigon Bay (Vietnamese), 2373 E Fowler Ave, Tampa, FL 33612
• Taj Indian Cuisine, 2734 E Fowler Ave, Tampa, FL 33612
• Ichiban Japanese Cuisine & Sushi Bar, 2786 E Fowler Ave, Tampa, FL 33612
• Carrabba’s Italian Grill, 5515 E Fowler Ave, Temple Terrace, FL 33617
• Wood Fired (Pizza), 2822 E Bearss Ave, Tampa, FL 33613
• Mr. Dunderbak’s Biergarten and Brewery, 14929 Bruce B Downs Blvd, Tampa,FL 33613 (driving needed)
• Acropolis Greek Taverna, 14947 Bruce B Downs Blvd, Tampa, FL 33613(driving needed)
• Thai Ruby, 15319 Amberly Dr, Tampa, FL 33647 (driving needed)
• Koizi Endless Hibachi & Sushi Eatery, 17012 Palm Pointe Dr, Tampa, FL33647 (driving needed)
• Petra Restaurant(Middle Eastern Cuisine), 4812 E Busch Blvd # E, Tampa,FL 33617 (driving needed)
Some Places of Interest (with no intent to be exhaustive):
• Lettuce Lake Park, 6920 E Fletcher Ave, Tampa, FL 33637 ($2 per car). Avery nice nature park featuring boardwalks over the lake. A place to seealligators and other wildlife. Not far from campus, but driving is needed.
• The Dali Museum, 1 Dali Blvd, St. Petersburg, FL 33701. About 40 minutedrive from Campus. It houses the largest collection of Dalı’s works outsideEurope.
• John and Mable Ringling Museum of Art. The most celebrated items inthe museum are 16th–20th-century European paintings, including a world-renowned collection of Peter Paul Rubens paintings. About 1 hour drive fromcampus.
• Clearwater Beach, Sarasota Beach (more touristy), and Sand Key Beach (lesstouristy). About 50 minute drive from campus.
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2018 Zassenhaus Group Theory and Friends ConferenceUniversity of South Florida, Tampa, FL
April 6–8, 2018
Abstracts
In order of presentation
A counterexample to the first Zassenhaus conjecture
Leo Margolis, Free University of Brussels, Belgium
Zassenhaus conjectured in 1974 that any unit of finite order in the integral groupring of a finite group G is conjugate in the rational group algebra of G to an elementof G, possibly up to sign.
I will recall some history of the problem and then present a recently foundmetabelian counterexample. The existence of the counterexample is equivalent toshowing the existence of a certain module over an integral group ring, which can beachieved by showing first the existence of certain modules over p-adic group ringsand then considering the genus class group. These general arguments allow to boildown the question to elementary character and group theoretic questions.
This is joint work with Florian Eisele.
A result on the Chermak-Delgado lattice of a finite group
Ryan McCulloch, University of Bridgeport
The Chermak-Delgado lattice of a finite group G, denoted CD(G), is a modu-lar, self-dual subgroup lattice, which has many nice properties. It is still an openquestion to characterize groups G for which CD(G) is a single point.
In this talk we sketch a proof of the following theorem:
Let G = AB be a finite group where A and B are abelian, A and B areof coprime order, and A is normal in G. Then CD(G) = ACB(A).
The proof uses a variant on Brodkey’s Theorem on Sylow intersections. This isjoint work with Marius Tarnauceanu of A.I. Cuza University, Iasi, Romania.
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On integers that are covering numbers of groups
Luise-Charlotte Kappe, Binghamton University
If a group G is the union of proper subgroups H1, . . . ,Hk, we say that thecollection H1, . . . ,Hk is a cover of G and the size of a minimal cover (supposingone exists) is the covering number of G, denoted by σ(G). The authors determinedall integers less than 130 that are covering numbers, in addition to generalizing aresult of Tomkinson and showing that every integer of the form qn−1
q−1, where q is a
prime power and n 6= w, is a cover number. These results will be discussed duringthe talk, in addition to a discussion of the progress made towards proving that thereare infinitely many integers that are not covering numbers of groups.
Two new criteria for solvability of finite groups
Patrizia Longobardi, Universita di Salerno, Italy
The aim of my talk is to present two sufficient conditions for a finite group tobe solvable, obtained jointly with Marcel Herzog and Mercede Maj.
A characterization of solvable A-groups and its generalization to
universal algebra
Eran Crockett, Binghamton University
An A-group is a finite group in which all Sylow subgroups are abelian. We findtwo characterizations of solvable A-groups that do not depend on Sylow subgroups.With the knowledge that solvable A-groups are the finite solvable groups that avoidnon-abelian nilpotence, we attempt to characterize the finite nilpotent algebras thatavoid non-abelian supernilpotence.
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Smith and critical groups of polar graphs
Venkata Raghu Tej Pantangi, University of Florida
The Smith and critical groups of a graph are interesting invariants. The Smithgroup of a graph is the abelian group whose cyclic decomposition is given by theSmith normal form of the adjacency matrix of the graph. The critical group is thefinite part of the abelian group whose cyclic decomposition is given by the Smithnormal form of the Laplacian matrix of a graph. An active line of research has beento calculate the Smith and critical groups of families of strongly regular graphs. Inthis presentation, we shall compute these groups for families of Polar graphs. Theseare strongly regular graphs associated with the rank 3 permutation action of thesome finite classical groups. This is joint work with Peter Sin.
The Mobius function of the affine linear group AGL(1,Fq)Xiang-dong Hou, University of South Florida
The Mobius function of a finite group is the Mobius function of the lattice ofsubgroups of the group. The Mobius function is an important tool for studying thestructure of the group and its actions on other structures. However, the Mobiusfunction is known only for a few classes of finite groups. In this talk we describe a re-cent work that determines the Mobius function of the affine linear group AGL(1,Fq)over a finite field.
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The Benson-Solomon fusion systems
Justin Lynd, University of Louisiana at Lafayette
Given a finite group G and a prime p, one can form the fusion system of G atp. This is a category whose objects are the subgroups of a fixed Sylow p-subgroupS, and where the morphisms are the conjugation homomorphisms induced by theelements of G. The notion of a saturated fusion system is abstracted from thisstandard example, and provides a coarse representation of what is meant by “ap-local structure” of a finite group. Once the group G is abstracted away, thereappear many exotic fusion systems not arising in the above fashion. Exotic fusionsystems are prevalent at odd primes, but only a single one-parameter family of“simple” fusion systems at the prime 2 are currently known. These are closelyrelated to the groups Spin7(q), q odd, and were first considered by Solomon andBenson, although not as fusion systems per se. I’ll explain some of the coincidencesthat allow the Benson-Solomon systems to exist, and then discuss various resultsabout these systems as time allows. This may include a description of their outerautomorphism groups (joint with E. Henke), the number of simple modules thesesystems would have if they arose from blocks of group algebras in characteristic 2(with J. Semeraro), as well as fusion systems at the prime 2 in which a Benson-Solomon system is subnormal in the centralizer of an involution (with E. Henke).
Rational class sizes and their implications about the structure of a
finite group
Hossein Shahrtash, University of Florida
Ever since Ito introduced the notion of a conjugate type vector in 1953, theproblem of unraveling the connections between the set of conjugacy class sizes andthe structure of a finite group has been widely studied. There are interesting in-stances of recognizing structural properties of a finite group, including solvability,nilpotency, etc. based on the set of conjugacy class sizes. In this talk, we will lookat a problem of similar nature by considering the sizes of rational classes in a finitegroup. Knowing the sizes of rational classes in a finite group, how much informationcan we expect to obtain about the structural properties of the group?
16
Bounds of nilpotency class of finite p-groupsRisto Atanasov, Western Carolina University
A finite p-group G is called powerful if either p is odd and [G,G] ⊆ Gp or p = 2and [G,G] ⊆ G4. We will discuss results that bound the nilpotency class of apowerful p-group and a p-central group in terms of the exponent of a quotient bya normal abelian subgroup. This is a joint work with Ilir Snopche and SlobodanTanushevski.
Matrix loops over proper Kalscheuer near-fields
Clifton E. Ealy, Jr., Western Michigan University
Let K have addition, multiplication, and distinct multiplicative, 1, and additive,0, identities. Informally, K is a near-domain if K additively is a Bol loop withautomorphic inverse property, multiplicatively K/0 is a group, and one of the thedistributive laws hold. Every field is a near-field, every near-field is a near-domain;but, not vice versa. In this talk we will introduce nxn loops over a proper Kalscheuernear-field. (Key words: Sharply 2-transitive groups, loops, Dieudonne determinate,near-fields.)
An Euler totient sum inequality
Brian Curtin, University of South Florida
The power graph of a finite group is the undirected graph whose vertices are thegroup elements and two elements are adjacent if one is a power of the other. Weshow that the clique number of the power graph of a cyclic group is given by a nicefunction of elementary number theory. We discuss some properties of this function.
17
On the Burghelea conjecture
Alexander Dranishnikov, University of Florida
The Burghelea conjecture is a conjecture about groups stated in terms of cyclichomology. It implies the Idempotent conjecture. It is verified for some classes ofgroups. Genelly it is false. We will discuss the Burghelea conjecture for groups withfinite macrscopic dimension.
Distributive groupoids and their cohomologies
Emanuele Zappala, University of South Florida
Conjugation in a group defines a self distributive binary operation. A distribu-tive groupoid (aka quandle) is an algebraic structure that generalizes the notion ofconjugation operation in a group. In this talk I will introduce the notion of coho-mology of quandles and investigate some of the main features: as for cohomologyof groups, the second cohomology of a quandle is in bijective correspondence withthe extensions of the quandle. I will also introduce the notion of inverse limit ofquandles and discuss its cohomology.
On groups with countably many maximal subgroups
Derek Robinson, University of Illinois at Urbana-Champaign
We will describe classes of groups which have only countably many maximalsubgroups, and also give examples of finitely generated soluble groups with un-countably many maximal subgroups. Connections with rings that have countablymany maximal right ideals and modules with countably many maximal submoduleswill also be discussed.
18
Invariant random subgroups and factor representations of branch
groups
Rostislav Grigorchuk, Texas A&M University
I will discuss various notions of non-free action and measure spaces and why theyare useful for study invariant random (IRS) subgroups and factor representations.This will be applied to branch groups to show that they have uncountably manyergodic continuous IRS’s and factor representations of type II1.
On growth of generalized Grigorchuk’s overgroups
Supun Samarakoon, Texas A&M University
Grigorchuk’s Overgroup G, first described in Bartholdi and Grigorchuk’s 1999paper, “On the spectrum of Hecke type operators related to some fractal groups”and revisited in their 2001 paper, “On parabolic subgroups and Hecke algebras ofsome fractal groups,” is a branch group of intermediate growth. It contains the firstGrigorchuk’s torsion group G of intermediate growth constructed by Grigorchuk in1980, but also has elements of infinite order. Its growth is substantially greaterthan the growth of G. The group G, corresponding to the sequence 012012 . . ., isa member of the family Gω, ω ∈ Ω = 0, 1, 2N , as proved in Grigorchuk’s 1984paper, consisting of groups of intermediate growth when sequence ω is not virtuallyconstant. Following the construction from 1984, we define generalized overgroupsGω, ω ∈ Ω such that Gω is a subgroup of Gω for each ω ∈ Ω. We prove,
• If ω is eventually constant, then Gω is of polynomial growth and hence virtu-ally abelian.
• If ω is not eventually constant, then Gω is of intermediate growth.
19
Simple groups of intermediate growth
Volodymyr Nekrashevych, Texas A&M University
We will discuss several examples of simple groups of intermediate growth: agroup associated with the golden ratio rotation, a group containing the Grigorchukgroup, a group acting on the Thue-Morse subshift.
Enumerating periodic graphs
Gregory McColm, University of South Florida
A graph Γ is periodic if, for some finite dimensional free abelian L ≤ Aut(Γ),Γ/L is finite. A d-dimensional periodic graph is realized by a geometric graph πΓ ind-dimensional Euclidean space such that πΓ ∼= Γ; typically, we are only interested inrealizations for which πL is a group of translational symmetries of πΓ. We outlinea project to effectively enumerate geometric realizations of periodic graphs withingiven restrictions. This is a project with applications in crystal engineering.
Finite-state automata and measures
Roman Kogan, Industry / Former Texas A&M University
The idea of self-similarity has been prominently used in group theory ever sincethe introduction of the Grigorchuk group, generated by states of a finite-state ma-chine with output, to answer Milnor’s question on intermediate growth of groups.Similar ideas can be applied to the study of measures on the space of sequences in afinite alphabet to define finite-state measures. These measures generalize Bernoulli,Markov and k-step Markov measures in a natural way, and are preserved by theaction of invertible finite-state automorphisms. We introduce and briefly discussthe properties of these measures, such as when they are k-step Markov, and whentheir image under non-invertible automorphisms is finite-state.
20
Rewriting in Thompson’s group FZoran Sunic, Hofstra University
It is not known if Thompson’s group F admits a finite confluent rewriting system.We construct a system that is not finite, “but it comes close.” Namely, we constructa regular, bounded, prefix-rewriting system for F over its standard 2-generator set.Modulo the jargon, this means that one can rewrite any word to its normal form,and thus solve the word problem, by using a device with uniformly bounded amountof memory – in other words, even I can do it. Our system is based on the rewritingsystem and the corresponding normal form introduced by Victor Guba and MarkSapir in 1997.
Mathematical models for describing molecular self-assembly
Margherita Maria Ferrari, University of South Florida
We present several mathematical models for describing molecular building blocks,called rigid tiles, that assemble in larger nanostructures. Rigid tiles can be seen ask-arm branch junction structures that join together by annealing to each otherthrough the affinity of their arm-ends. Such a k-arm rigid tile is described with kvectors joined at the origin that can be translated or rotated during the assembly.Besides the geometric shape of the building blocks, the models can take into accountthe geometry of the arm-ends joining together. We show distinctions between fourmodels by characterizing types of structures that can be assembled and we outlinean algebraic approach to characterize nanostructures built by a set of rigid tiles.
Quandles and groups
Mohamed Elhamdadi, University of South Florida
Quandles are non-associative algebraic structures whose motivation comes par-tially from the study of knot theory. We will give a survey of these structuresfocusing more on the algebraic side.
21
The game GENERATE on finite nilpotent groups
Bret Benesh, College of St. Benedict and St. John’s University
We study an impartial game introduced by Anderson and Harary. The game isplayed by two players who alternately choose previously-unselected elements of afinite group. The first player who builds a generating set from the jointly-selectedelements wins.
We determine the nim-numbers of this game for finite groups with a 2-Sylowdirect factor, that is the group is of the form T ×H, where T is a 2-group and H isa group of odd order. This includes all nilpotent and hence abelian groups.
Endomorphisms of regular rooted trees induced by the action of
polynomials on the ring Zd of d-adic integers
Elsayed Ahmed, University of South Florida
We show that every polynomial in Z[x] defines an endomorphism of the d-aryrooted tree induced by its action on the ring Zd of d-adic integers. The sectionsof this endomorphism also turn out to be induced by polynomials in Z[x] of thesame degree. In the case of permutational polynomials acting on Zd by bijectionsthe induced endomorphisms are automorphisms of the tree. In the case of Z2 suchpolynomials were completely characterized by Rivest. As our main application weutilize the result of Rivest to derive the condition on the coefficients of a permuta-tional polynomial f(x) ∈ Z[x] that is necessary and sufficient for f to induce a leveltransitive automorphism of the binary tree, which is equivalent to the ergodicity ofthe action of f(x) on Z2 with respect to the normalized Haar measure.
On Liouville property of action of discrete groups
Kate Juschenko, Northwestern University
We will discuss Liouville property of actions, relate it to amenability and toadditive combinations for certain classes of groups.
22
Groups whose non-permutable subgroups are soluble minimax
Zekeriya Karatas, University of Cincinnati Blue Ash College
Determining the structure of groups whose proper subgroups satisfy certainconditions has been a very well-known problem in group theory. Many interestingresults have been found through the history of this area. In this talk, the structureof locally graded groups whose non-permutable subgroups satisfy certain conditionswill be given. In particular, I will conclude with the structure of groups whosesubgroups are permutable or soluble minimax. I will give the history of thesetype of problems including the most significant results, definitions, and some openproblems.
The invariant of a character
Alexandre Turull, University of Florida
Let p be a prime. We assume, as is customary, that we can assign p-Brauercharacters to modules of finite groups in characteristic p. The elements of theBrauer group of any finite extension of Qp, the field of p-adic numbers, are innatural one to one correspondence with their corresponding invariants in Q/Z. LetG be a finite group, and let χ be an irreducible character of G. We assume, as iscustomary, that χ has complex values. Even though χ may not correspond to aunique irreducible character of G with coefficients in some algebraic closure of Qp,the character χ nevertheless determines a unique invariant in Q/Z.
On the rigidity of rank gradient in a group of intermediate growth
Rostyslav Kravchenko, Northwestern University
We introduce and investigate the rigidity property of rank gradient in the caseof the Grigorchuk group. We show that it it normally (log, log log)-RG rigid. Thisis a joint work with R. Grigorchuk.
23
A generalized version of nilpotence arising from supercharacter theory
Shawn Burkett, University of Colorado Boulder
Since its introduction, supercharacter theory has been used to study a widevariety of problems. However, the structure of supercharacter theories themselvesremains mysterious. In this talk, we will discuss supercharacter theories that areable to detect nilpotence, in some sense. By defining analogs of the center and com-mutator subgroup for a given supercharacter theory S of G, one may use these todefine a coarser version of nilpotence, which we call S- nilpotence. The superchar-acter theories S of a nilpotent group G for which G is S-nilpotent will be classified,with particular emphasis on p-groups. Then some potential applications and furthergeneralizations will be discussed, as time permits.
Proving Turing universality of cotranscriptional folding
Shinnosuke Seki, University of Electro-Communications, Tokyo
Transcription is a process in which an RNA sequence (of letter A, C, G, U)is synthesized out of a template DNA sequence (of A, C, G, T ) according to therule A → U , C → G, G → C, and T → A by an RNA polymerase enzyme. Theelongating (incomplete) RNA sequence (transcript) starts folding upon itself viahydrogen bonds into a stable tertiary conformation. Cotranscriptional folding refersto this phenomenon. Cotranscriptional folding plays various roles in informationprocessing in organisms such as regulation of gene expression and splicing. Usingoritatami system, the novel mathematical model of cotranscriptional folding, weprove the Turing universality of oritatami system, which implies the capability ofcotranscriptional folding for computing an arbitrary computable function.
24
2018 Zassenhaus Group Theory and Friends ConferenceUniversity of South Florida, Tampa, FL
April 6–8, 2018
Abstracts
Alphabetical by Speaker
25
Endomorphisms of regular rooted trees induced by the action of
polynomials on the ring Zd of d-adic integers
Elsayed Ahmed, University of South Florida
We show that every polynomial in Z[x] defines an endomorphism of the d-aryrooted tree induced by its action on the ring Zd of d-adic integers. The sectionsof this endomorphism also turn out to be induced by polynomials in Z[x] of thesame degree. In the case of permutational polynomials acting on Zd by bijectionsthe induced endomorphisms are automorphisms of the tree. In the case of Z2 suchpolynomials were completely characterized by Rivest. As our main application weutilize the result of Rivest to derive the condition on the coefficients of a permuta-tional polynomial f(x) ∈ Z[x] that is necessary and sufficient for f to induce a leveltransitive automorphism of the binary tree, which is equivalent to the ergodicity ofthe action of f(x) on Z2 with respect to the normalized Haar measure.
Bounds of nilpotency class of finite p-groupsRisto Atanasov, Western Carolina University
A finite p-group G is called powerful if either p is odd and [G,G] ⊆ Gp or p = 2and [G,G] ⊆ G4. We will discuss results that bound the nilpotency class of apowerful p-group and a p-central group in terms of the exponent of a quotient bya normal abelian subgroup. This is a joint work with Ilir Snopche and SlobodanTanushevski.
27
The game GENERATE on finite nilpotent groups
Bret Benesh, College of St. Benedict and St. John’s University
We study an impartial game introduced by Anderson and Harary. The game isplayed by two players who alternately choose previously-unselected elements of afinite group. The first player who builds a generating set from the jointly-selectedelements wins.
We determine the nim-numbers of this game for finite groups with a 2-Sylowdirect factor, that is the group is of the form T ×H, where T is a 2-group and H isa group of odd order. This includes all nilpotent and hence abelian groups.
A generalized version of nilpotence arising from supercharacter theory
Shawn Burkett, University of Colorado Boulder
Since its introduction, supercharacter theory has been used to study a widevariety of problems. However, the structure of supercharacter theories themselvesremains mysterious. In this talk, we will discuss supercharacter theories that areable to detect nilpotence, in some sense. By defining analogs of the center and com-mutator subgroup for a given supercharacter theory S of G, one may use these todefine a coarser version of nilpotence, which we call S- nilpotence. The superchar-acter theories S of a nilpotent group G for which G is S-nilpotent will be classified,with particular emphasis on p-groups. Then some potential applications and furthergeneralizations will be discussed, as time permits.
28
A characterization of solvable A-groups and its generalization to
universal algebra
Eran Crockett, Binghamton University
An A-group is a finite group in which all Sylow subgroups are abelian. We findtwo characterizations of solvable A-groups that do not depend on Sylow subgroups.With the knowledge that solvable A-groups are the finite solvable groups that avoidnon-abelian nilpotence, we attempt to characterize the finite nilpotent algebras thatavoid non-abelian supernilpotence.
An Euler totient sum inequality
Brian Curtin, University of South Florida
The power graph of a finite group is the undirected graph whose vertices are thegroup elements and two elements are adjacent if one is a power of the other. Weshow that the clique number of the power graph of a cyclic group is given by a nicefunction of elementary number theory. We discuss some properties of this function.
On the Burghelea conjecture
Alexander Dranishnikov, University of Florida
The Burghelea conjecture is a conjecture about groups stated in terms of cyclichomology. It implies the Idempotent conjecture. It is verified for some classes ofgroups. Genelly it is false. We will discuss the Burghelea conjecture for groups withfinite macrscopic dimension.
29
Matrix loops over proper Kalscheuer near-fields
Clifton E. Ealy, Jr., Western Michigan University
Let K have addition, multiplication, and distinct multiplicative, 1, and additive,0, identities. Informally, K is a near-domain if K additively is a Bol loop withautomorphic inverse property, multiplicatively K/0 is a group, and one of the thedistributive laws hold. Every field is a near-field, every near-field is a near-domain;but, not vice versa. In this talk we will introduce nxn loops over a proper Kalscheuernear-field. (Key words: Sharply 2-transitive groups, loops, Dieudonne determinate,near-fields.)
Quandles and groups
Mohamed Elhamdadi, University of South Florida
Quandles are non-associative algebraic structures whose motivation comes par-tially from the study of knot theory. We will give a survey of these structuresfocusing more on the algebraic side.
Mathematical models for describing molecular self-assembly
Margherita Maria Ferrari, University of South Florida
We present several mathematical models for describing molecular building blocks,called rigid tiles, that assemble in larger nanostructures. Rigid tiles can be seen ask-arm branch junction structures that join together by annealing to each otherthrough the affinity of their arm-ends. Such a k-arm rigid tile is described with kvectors joined at the origin that can be translated or rotated during the assembly.Besides the geometric shape of the building blocks, the models can take into accountthe geometry of the arm-ends joining together. We show distinctions between fourmodels by characterizing types of structures that can be assembled and we outlinean algebraic approach to characterize nanostructures built by a set of rigid tiles.
30
Invariant random subgroups and factor representations of branch
groups
Rostislav Grigorchuk, Texas A&M University
I will discuss various notions of non-free action and measure spaces and why theyare useful for study invariant random (IRS) subgroups and factor representations.This will be applied to branch groups to show that they have uncountably manyergodic continuous IRS’s and factor representations of type II1.
The Mobius function of the affine linear group AGL(1,Fq)Xiang-dong Hou, University of South Florida
The Mobius function of a finite group is the Mobius function of the lattice ofsubgroups of the group. The Mobius function is an important tool for studying thestructure of the group and its actions on other structures. However, the Mobiusfunction is known only for a few classes of finite groups. In this talk we describe a re-cent work that determines the Mobius function of the affine linear group AGL(1,Fq)over a finite field.
On Liouville property of action of discrete groups
Kate Juschenko, Northwestern University
We will discuss Liouville property of actions, relate it to amenability and toadditive combinations for certain classes of groups.
31
On integers that are covering numbers of groups
Luise-Charlotte Kappe, Binghamton University
If a group G is the union of proper subgroups H1, . . . ,Hk, we say that thecollection H1, . . . ,Hk is a cover of G and the size of a minimal cover (supposingone exists) is the covering number of G, denoted by σ(G). The authors determinedall integers less than 130 that are covering numbers, in addition to generalizing aresult of Tomkinson and showing that every integer of the form qn−1
q−1, where q is a
prime power and n 6= w, is a cover number. These results will be discussed duringthe talk, in addition to a discussion of the progress made towards proving that thereare infinitely many integers that are not covering numbers of groups.
Groups whose non-permutable subgroups are soluble minimax
Zekeriya Karatas, University of Cincinnati Blue Ash College
Determining the structure of groups whose proper subgroups satisfy certainconditions has been a very well-known problem in group theory. Many interestingresults have been found through the history of this area. In this talk, the structureof locally graded groups whose non-permutable subgroups satisfy certain conditionswill be given. In particular, I will conclude with the structure of groups whosesubgroups are permutable or soluble minimax. I will give the history of thesetype of problems including the most significant results, definitions, and some openproblems.
32
Finite-state automata and measures
Roman Kogan, Industry / Former Texas A&M University
The idea of self-similarity has been prominently used in group theory ever sincethe introduction of the Grigorchuk group, generated by states of a finite-state ma-chine with output, to answer Milnor’s question on intermediate growth of groups.Similar ideas can be applied to the study of measures on the space of sequences in afinite alphabet to define finite-state measures. These measures generalize Bernoulli,Markov and k-step Markov measures in a natural way, and are preserved by theaction of invertible finite-state automorphisms. We introduce and briefly discussthe properties of these measures, such as when they are k-step Markov, and whentheir image under non-invertible automorphisms is finite-state.
On the rigidity of rank gradient in a group of intermediate growth
Rostyslav Kravchenko, Northwestern University
We introduce and investigate the rigidity property of rank gradient in the caseof the Grigorchuk group. We show that it it normally (log, log log)-RG rigid. Thisis a joint work with R. Grigorchuk.
Two new criteria for solvability of finite groups
Patrizia Longobardi, Universita di Salerno, Italy
The aim of my talk is to present two sufficient conditions for a finite group tobe solvable, obtained jointly with Marcel Herzog and Mercede Maj.
33
The Benson-Solomon fusion systems
Justin Lynd, University of Louisiana at Lafayette
Given a finite group G and a prime p, one can form the fusion system of G atp. This is a category whose objects are the subgroups of a fixed Sylow p-subgroupS, and where the morphisms are the conjugation homomorphisms induced by theelements of G. The notion of a saturated fusion system is abstracted from thisstandard example, and provides a coarse representation of what is meant by “ap-local structure” of a finite group. Once the group G is abstracted away, thereappear many exotic fusion systems not arising in the above fashion. Exotic fusionsystems are prevalent at odd primes, but only a single one-parameter family of“simple” fusion systems at the prime 2 are currently known. These are closelyrelated to the groups Spin7(q), q odd, and were first considered by Solomon andBenson, although not as fusion systems per se. I’ll explain some of the coincidencesthat allow the Benson-Solomon systems to exist, and then discuss various resultsabout these systems as time allows. This may include a description of their outerautomorphism groups (joint with E. Henke), the number of simple modules thesesystems would have if they arose from blocks of group algebras in characteristic 2(with J. Semeraro), as well as fusion systems at the prime 2 in which a Benson-Solomon system is subnormal in the centralizer of an involution (with E. Henke).
A counterexample to the first Zassenhaus conjecture
Leo Margolis, Free University of Brussels, Belgium
Zassenhaus conjectured in 1974 that any unit of finite order in the integral groupring of a finite group G is conjugate in the rational group algebra of G to an elementof G, possibly up to sign.
I will recall some history of the problem and then present a recently foundmetabelian counterexample. The existence of the counterexample is equivalent toshowing the existence of a certain module over an integral group ring, which can beachieved by showing first the existence of certain modules over p-adic group ringsand then considering the genus class group. These general arguments allow to boildown the question to elementary character and group theoretic questions.
This is joint work with Florian Eisele.
34
Enumerating periodic graphs
Gregory McColm, University of South Florida
A graph Γ is periodic if, for some finite dimensional free abelian L ≤ Aut(Γ),Γ/L is finite. A d-dimensional periodic graph is realized by a geometric graph πΓ ind-dimensional Euclidean space such that πΓ ∼= Γ; typically, we are only interested inrealizations for which πL is a group of translational symmetries of πΓ. We outlinea project to effectively enumerate geometric realizations of periodic graphs withingiven restrictions. This is a project with applications in crystal engineering.
A result on the Chermak-Delgado lattice of a finite group
Ryan McCulloch, University of Bridgeport
The Chermak-Delgado lattice of a finite group G, denoted CD(G), is a modu-lar, self-dual subgroup lattice, which has many nice properties. It is still an openquestion to characterize groups G for which CD(G) is a single point.
In this talk we sketch a proof of the following theorem:
Let G = AB be a finite group where A and B are abelian, A and B areof coprime order, and A is normal in G. Then CD(G) = ACB(A).
The proof uses a variant on Brodkey’s Theorem on Sylow intersections. This isjoint work with Marius Tarnauceanu of A.I. Cuza University, Iasi, Romania.
Simple groups of intermediate growth
Volodymyr Nekrashevych, Texas A&M University
We will discuss several examples of simple groups of intermediate growth: agroup associated with the golden ratio rotation, a group containing the Grigorchukgroup, a group acting on the Thue-Morse subshift.
35
Smith and critical groups of polar graphs
Venkata Raghu Tej Pantangi, University of Florida
The Smith and critical groups of a graph are interesting invariants. The Smithgroup of a graph is the abelian group whose cyclic decomposition is given by theSmith normal form of the adjacency matrix of the graph. The critical group is thefinite part of the abelian group whose cyclic decomposition is given by the Smithnormal form of the Laplacian matrix of a graph. An active line of research has beento calculate the Smith and critical groups of families of strongly regular graphs. Inthis presentation, we shall compute these groups for families of Polar graphs. Theseare strongly regular graphs associated with the rank 3 permutation action of thesome finite classical groups. This is joint work with Peter Sin.
On groups with countably many maximal subgroups
Derek Robinson, University of Illinois at Urbana-Champaign
We will describe classes of groups which have only countably many maximalsubgroups, and also give examples of finitely generated soluble groups with un-countably many maximal subgroups. Connections with rings that have countablymany maximal right ideals and modules with countably many maximal submoduleswill also be discussed.
36
On growth of generalized Grigorchuk’s overgroups
Supun Samarakoon, Texas A&M University
Grigorchuk’s Overgroup G, first described in Bartholdi and Grigorchuk’s 1999paper, “On the spectrum of Hecke type operators related to some fractal groups”and revisited in their 2001 paper, “On parabolic subgroups and Hecke algebras ofsome fractal groups,” is a branch group of intermediate growth. It contains the firstGrigorchuk’s torsion group G of intermediate growth constructed by Grigorchuk in1980, but also has elements of infinite order. Its growth is substantially greaterthan the growth of G. The group G, corresponding to the sequence 012012 . . ., isa member of the family Gω, ω ∈ Ω = 0, 1, 2N , as proved in Grigorchuk’s 1984paper, consisting of groups of intermediate growth when sequence ω is not virtuallyconstant. Following the construction from 1984, we define generalized overgroupsGω, ω ∈ Ω such that Gω is a subgroup of Gω for each ω ∈ Ω. We prove,
• If ω is eventually constant, then Gω is of polynomial growth and hence virtu-ally abelian.
• If ω is not eventually constant, then Gω is of intermediate growth.
Proving Turing universality of cotranscriptional folding
Shinnosuke Seki, University of Electro-Communications, Tokyo
Transcription is a process in which an RNA sequence (of letter A, C, G, U)is synthesized out of a template DNA sequence (of A, C, G, T ) according to therule A → U , C → G, G → C, and T → A by an RNA polymerase enzyme. Theelongating (incomplete) RNA sequence (transcript) starts folding upon itself viahydrogen bonds into a stable tertiary conformation. Cotranscriptional folding refersto this phenomenon. Cotranscriptional folding plays various roles in informationprocessing in organisms such as regulation of gene expression and splicing. Usingoritatami system, the novel mathematical model of cotranscriptional folding, weprove the Turing universality of oritatami system, which implies the capability ofcotranscriptional folding for computing an arbitrary computable function.
37
Rational class sizes and their implications about the structure of a
finite group
Hossein Shahrtash, University of Florida
Ever since Ito introduced the notion of a conjugate type vector in 1953, theproblem of unraveling the connections between the set of conjugacy class sizes andthe structure of a finite group has been widely studied. There are interesting in-stances of recognizing structural properties of a finite group, including solvability,nilpotency, etc. based on the set of conjugacy class sizes. In this talk, we will lookat a problem of similar nature by considering the sizes of rational classes in a finitegroup. Knowing the sizes of rational classes in a finite group, how much informationcan we expect to obtain about the structural properties of the group?
Rewriting in Thompson’s group FZoran Sunic, Hofstra University
It is not known if Thompson’s group F admits a finite confluent rewriting system.We construct a system that is not finite, “but it comes close.” Namely, we constructa regular, bounded, prefix-rewriting system for F over its standard 2-generator set.Modulo the jargon, this means that one can rewrite any word to its normal form,and thus solve the word problem, by using a device with uniformly bounded amountof memory – in other words, even I can do it. Our system is based on the rewritingsystem and the corresponding normal form introduced by Victor Guba and MarkSapir in 1997.
38
The invariant of a character
Alexandre Turull, University of Florida
Let p be a prime. We assume, as is customary, that we can assign p-Brauercharacters to modules of finite groups in characteristic p. The elements of theBrauer group of any finite extension of Qp, the field of p-adic numbers, are innatural one to one correspondence with their corresponding invariants in Q/Z. LetG be a finite group, and let χ be an irreducible character of G. We assume, as iscustomary, that χ has complex values. Even though χ may not correspond to aunique irreducible character of G with coefficients in some algebraic closure of Qp,the character χ nevertheless determines a unique invariant in Q/Z.
Distributive groupoids and their cohomologies
Emanuele Zappala, University of South Florida
Conjugation in a group defines a self distributive binary operation. A distribu-tive groupoid (aka quandle) is an algebraic structure that generalizes the notion ofconjugation operation in a group. In this talk I will introduce the notion of coho-mology of quandles and investigate some of the main features: as for cohomologyof groups, the second cohomology of a quandle is in bijective correspondence withthe extensions of the quandle. I will also introduce the notion of inverse limit ofquandles and discuss its cohomology.
39
Notes
40
Notes
41