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# 2018 Zassenhaus Groups and Friends Conference — Abstracts of Talks

## Alphabetical by Speaker

Endomorphisms of regular rooted trees induced by the action of polynomials on the ring $$\mathbb{Z}_d$$ of $$d$$-adic integers
Elsayed Ahmed, University of South Florida

We show that every polynomial in $$\mathbb{Z}[x]$$ defines an endomorphism of the $$d$$-ary rooted tree induced by its action on the ring $$\mathbb{Z}_d$$ of $$d$$-adic integers. The sections of this endomorphism also turn out to be induced by polynomials in $$\mathbb{Z}[x]$$ of the same degree. In the case of permutational polynomials acting on Zd by bijections the induced endomorphisms are automorphisms of the tree. In the case of $$\mathbb{Z}_2$$ such polynomials were completely characterized by Rivest. As our main application we utilize the result of Rivest to derive the condition on the coefficients of a permutational polynomial $$f(x)\in\mathbb{Z}[x]$$ that is necessary and sufficient for $$f$$ to induce a level transitive automorphism of the binary tree, which is equivalent to the ergodicity of the action of $$f(x)$$ on $$\mathbb{Z}_2$$ with respect to the normalized Haar measure.

Bounds of nilpotency class of finite $$p$$-groups
Risto Atanasov, Western Carolina University

A finite $$p$$-group $$G$$ is called powerful if either $$p$$ is odd and $$[G,G]\subseteq G^p$$ or $$p=2$$ and $$[G,G]\subseteq G^4$$. We will discuss results that bound the nilpotency class of a powerful $$p$$-group and a $$p$$-central group in terms of the exponent of a quotient by a normal abelian subgroup. This is a joint work with Ilir Snopche and Slobodan Tanushevski.

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 is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins.

We determine the nim-numbers of this game for finite groups with a 2-Sylow direct factor, that is the group is of the form $$T\times H$$, where $$T$$ is a 2-group and $$H$$ is a 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 wide variety of problems. However, the structure of supercharacter theories themselves remains mysterious. In this talk, we will discuss supercharacter theories that are able to detect nilpotence, in some sense. By defining analogs of the center and commutator subgroup for a given supercharacter theory $$S$$ of $$G$$, one may use these to define a coarser version of nilpotence, which we call $$S$$-nilpotence. The supercharacter 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 further generalizations will be discussed, as time permits.

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 find two 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 avoid non-abelian nilpotence, we attempt to characterize the finite nilpotent algebras that avoid 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 the group elements and two elements are adjacent if one is a power of the other. We show that the clique number of the power graph of a cyclic group is given by a nice function 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 cyclic homology. It implies the Idempotent conjecture. It is verified for some classes of groups. Genelly it is false. We will discuss the Burghelea conjecture for groups with finite macrscopic dimension.

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 with automorphic inverse property, multiplicatively $$K/\{0\}$$ is a group, and one of the the distributive laws hold. Every field is a nearfield, every near-field is a near-domain; but, not vice versa. In this talk we will introduce nxn loops over a proper Kalscheuer near-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 partially from the study of knot theory. We will give a survey of these structures focusing 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 as $$k$$-arm branch junction structures that join together by annealing to each other through the affinity of their arm-ends. Such a $$k$$-arm rigid tile is described with $$k$$ vectors 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 account the geometry of the arm-ends joining together. We show distinctions between four models by characterizing types of structures that can be assembled and we outline an algebraic approach to characterize nanostructures built by a set of rigid tiles.

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 they are useful for study invariant random (IRS) subgroups and factor representations. This will be applied to branch groups to show that they have uncountably many ergodic continuous IRS’s and factor representations of type $$II_1$$.

The Möbius function of the affine linear group $$\mathrm{AGL}(1,\mathbb{F}_q)$$
Xiang-dong Hou, University of South Florida

The Möbius function of a finite group is the Möbius function of the lattice of subgroups of the group. The Möbius function is an important tool for studying the structure of the group and its actions on other structures. However, the Möbius function is known only for a few classes of finite groups. In this talk we describe a recent work that determines the Möbius function of the affine linear group $$\operatorname{AGL}\left(1,\mathbb{F}_q\right)$$ 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 to additive combinations for certain classes of groups.

On integers that are covering numbers of groups
Luise-Charlotte Kappe, Binghamton University

If a group $$G$$ is the union of proper subgroups $$H_1,\dotsc,H_k$$, we say that the collection $$\left\{H_1,\dotsc,H_k\right\}$$ is a cover of $$G$$ and the size of a minimal cover (supposing one exists) is the covering number of $$G$$, denoted by $$\sigma(G)$$. The authors determined all integers less than 130 that are covering numbers, in addition to generalizing a result of Tomkinson and showing that every integer of the form $$\frac{qn-1}{q-1}$$, where $$q$$ is a prime power and $$n\ne w$$, is a cover number. These results will be discussed during the talk, in addition to a discussion of the progress made towards proving that there are 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 certain conditions has been a very well-known problem in group theory. Many interesting results have been found through the history of this area. In this talk, the structure of locally graded groups whose non-permutable subgroups satisfy certain conditions will be given. In particular, I will conclude with the structure of groups whose subgroups are permutable or soluble minimax. I will give the history of these type of problems including the most significant results, definitions, and some open problems.

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 since the introduction of the Grigorchuk group, generated by states of a finite-state machine 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 a finite alphabet to define finite-state measures. These measures generalize Bernoulli, Markov and $$k$$-step Markov measures in a natural way, and are preserved by the action of invertible finite-state automorphisms. We introduce and briefly discuss the properties of these measures, such as when they are k-step Markov, and when their 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 case of the Grigorchuk group. We show that it it normally ($$\log, \log\log$$)-$$\mathrm{RG}$$ rigid. This is a joint work with R. Grigorchuk.

Two new criteria for solvability of finite groups
Patrizia Longobardi, Università di Salerno, Italy

The aim of my talk is to present two sufficient conditions for a finite group to be solvable, obtained jointly with Marcel Herzog and Mercede Maj.

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 $$G$$ at $$p$$. This is a category whose objects are the subgroups of a fixed Sylow $$p$$-subgroup $$S$$, and where the morphisms are the conjugation homomorphisms induced by the elements of $$G$$. The notion of a saturated fusion system is abstracted from this standard example, and provides a coarse representation of what is meant by “a $$p$$-local structure” of a finite group. Once the group $$G$$ is abstracted away, there appear many exotic fusion systems not arising in the above fashion. Exotic fusion systems 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 closely related to the groups $$\operatorname{Spin}_7(q)$$, $$q$$ odd, and were first considered by Solomon and Benson, although not as fusion systems per se. I’ll explain some of the coincidences that allow the Benson-Solomon systems to exist, and then discuss various results about these systems as time allows. This may include a description of their outer automorphism groups (joint with E. Henke), the number of simple modules these systems 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 group ring of a finite group $$G$$ is conjugate in the rational group algebra of $$G$$ to an element of $$G$$, possibly up to sign.

I will recall some history of the problem and then present a recently found metabelian counterexample. The existence of the counterexample is equivalent to showing the existence of a certain module over an integral group ring, which can be achieved by showing first the existence of certain modules over p-adic group rings and then considering the genus class group. These general arguments allow to boil down the question to elementary character and group theoretic questions.

Enumerating periodic graphs
Gregory McColm, University of South Florida

A graph $$\Gamma$$ is periodic if, for some finite dimensional free abelian $$L\le\operatorname{Aut}\,(\Gamma)$$, $$\Gamma/L$$ is finite. A $$d$$-dimensional periodic graph is realized by a geometric graph $$\pi\Gamma$$ in $$d$$-dimensional Euclidean space such that $$\pi\Gamma\cong\Gamma$$; typically, we are only interested in realizations for which $$\pi L$$ is a group of translational symmetries of $$\pi L$$. We outline a project to effectively enumerate geometric realizations of periodic graphs within given 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 $$\mathcal{CD}(G)$$, is a modular, self-dual subgroup lattice, which has many nice properties. It is still an open question to characterize groups $$G$$ for which $$\mathcal{CD}(G)$$ 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 $$B$$ are of coprime order, and $$A$$ is normal in
$$G$$. Then $$\mathcal{CD}(G)=AC_B(A)$$.

The proof uses a variant on Brodkey’s Theorem on Sylow intersections. This is joint work with Marius Tărnăuceanu of A.I. Cuza University, Iași, Romania.

Simple groups of intermediate growth
Volodymyr Nekrashevych, Texas A&M University

We will discuss several examples of simple groups of intermediate growth: a group associated with the golden ratio rotation, a group containing the Grigorchuk group, a group acting on the Thue-Morse subshift.

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 Smith group of a graph is the abelian group whose cyclic decomposition is given by the Smith normal form of the adjacency matrix of the graph. The critical group is the finite part of the abelian group whose cyclic decomposition is given by the Smith normal form of the Laplacian matrix of a graph. An active line of research has been to calculate the Smith and critical groups of families of strongly regular graphs. In this presentation, we shall compute these groups for families of Polar graphs. These are strongly regular graphs associated with the rank 3 permutation action of the some 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 maximal subgroups, and also give examples of finitely generated soluble groups with uncountably many maximal subgroups. Connections with rings that have countably many maximal right ideals and modules with countably many maximal submodules will also be discussed.

On growth of generalized Grigorchuk’s overgroups
Supun Samarakoon, Texas A&M University

Grigorchuk’s Overgroup $$\tilde{G}$$, first described in Bartholdi and Grigorchuk’s 1999 paper, “On the spectrum of Hecke type operators related to some fractal groups” and revisited in their 2001 paper, “On parabolic subgroups and Hecke algebras of some fractal groups,” is a branch group of intermediate growth. It contains the first Grigorchuk’s torsion group $$G$$ of intermediate growth constructed by Grigorchuk in 1980, but also has elements of infinite order. Its growth is substantially greater than the growth of $$G$$. The group $$G$$, corresponding to the sequence $$012012\dotsc$$, is a member of the family $$\tilde{G}_\omega$$, $$\omega\in\Omega=\{0,1,2\}^{\mathbb N}$$, as proved in Grigorchuk’s 1984 paper, consisting of groups of intermediate growth when sequence $$\omega$$ is not virtually constant. Following the construction from 1984, we define generalized overgroups $$\left\{\tilde{G}_\omega,\omega\in\Omega\right\}$$ such that $$G_\omega$$ is a subgroup of $$\tilde{G}_\omega$$ for each $$\omega\in\Omega$$. We prove,

• If $$\omega$$ is eventually constant, then $$\tilde{G}_\omega$$ is of polynomial growth and hence virtually abelian.
• If $$\omega$$ is not eventually constant, then $$\tilde{G}_\omega$$ 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 the rule $$A\to U$$, $$C\to G$$, $$G\to C$$, and $$T\to A$$ by an RNA polymerase enzyme. The elongating (incomplete) RNA sequence (transcript) starts folding upon itself via hydrogen bonds into a stable tertiary conformation. Cotranscriptional folding refers to this phenomenon. Cotranscriptional folding plays various roles in information processing in organisms such as regulation of gene expression and splicing. Using oritatami system, the novel mathematical model of cotranscriptional folding, we prove the Turing universality of oritatami system, which implies the capability of cotranscriptional folding for computing an arbitrary computable function.

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, the problem of unraveling the connections between the set of conjugacy class sizes and the structure of a finite group has been widely studied. There are interesting instances 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 look at a problem of similar nature by considering the sizes of rational classes in a finite group. Knowing the sizes of rational classes in a finite group, how much information can we expect to obtain about the structural properties of the group?

Rewriting in Thompson’s group $$F$$
Zoran Šunić, 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 construct a 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 amount of memory — in other words, even I can do it. Our system is based on the rewriting system and the corresponding normal form introduced by Victor Guba and Mark Sapir in 1997.

The invariant of a character
Alexandre Turull, University of Florida

Let $$p$$ be a prime. We assume, as is customary, that we can assign $$p$$-Brauer characters to modules of finite groups in characteristic $$p$$. The elements of the Brauer group of any finite extension of $$\mathbf{Q}_p$$, the field of $$p$$-adic numbers, are in natural one to one correspondence with their corresponding invariants in $$\mathbf{Q}/\mathbf{Z}$$. Let $$G$$ be a finite group, and let $$\chi$$ be an irreducible character of $$G$$. We assume, as is customary, that $$\chi$$ has complex values. Even though $$\chi$$ may not correspond to a unique irreducible character of $$G$$ with coefficients in some algebraic closure of $$\mathbf{Q}_p$$, the character $$\chi$$ nevertheless determines a unique invariant in $$\mathbf{Q}/\mathbf{Z}$$.

Distributive groupoids and their cohomologies
Emanuele Zappala, University of South Florida

Conjugation in a group defines a self distributive binary operation. A distributive groupoid (aka quandle) is an algebraic structure that generalizes the notion of conjugation operation in a group. In this talk I will introduce the notion of cohomology of quandles and investigate some of the main features: as for cohomology of groups, the second cohomology of a quandle is in bijective correspondence with the extensions of the quandle. I will also introduce the notion of inverse limit of quandles and discuss its cohomology.