zassenhaus:zassenhaus_2025:programtest [Department of Mathematics and Statistics, Binghamton University]

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<body>
    <header>
        <nav>
            <div class="title">
              <h1>Zassenhaus Groups and Friends Conference 2025</h1>
                <p>May 31 - June 1, 2025</p>
            </div>
        </nav>
    </header>
    <main>

<section class="timetable" id="schedule">
<h2>Conference Schedule</h2>
<p>
  A detailed program (in PDF format) is
  available <a href="full-program.pdf">here</a><br>
  Clicking on a name will take you to the 
  talk's title below, and clicking on the title will display its abstract.
  
<table>
  <tbody><tr>
      <th>Time</th>
      <th>Saturday 5/31</th>
      <th>Sunday 6/1</th>
</tr>
<tr>
  <td>8:00</td>
  <td class="break" rowspan="2"> Registration</td>
  <td></td>
</tr>
<tr>
  <td>8:30</td>
  <td><a href="#Riedl"
	 style="color:black"><b>Riedl</b></a></td>
  </tr>
<tr>
    <td>9:00</td>
  <td><a href="#Guzman" style="color:black"><b>Guzman</b></a></td>
    <td><a href="#McCulloch"
	   style="color:black"><b>McCulloch</b></a></td>
</tr>

<tr>
  <td>9:30</td>
  <td><a href="#Donoven" style="color:black"><b>Donoven</b></a></td>
  <td><a href="#Cinarci" style="color:black"><b>&Ccedil;&imath;narc&imath;</b></a></td>
</tr>

    <tr>
  <td>10:00</td>
  <td><a href="#Tran" style="color:black"><b>Tran</b></a></td>
  <td class="break">Coffee Break</td>
</tr>

    <tr>
      <td>10:30</td>
      <td class="break">Coffee Break</td>
      <td><a href="#Zarrin" style="color:black"><b>Zarrin</b></a></td>
    </tr>

    <tr>
      <td>11:00</td>
      <td><a href="#Lewis" style="color:black"><b>Lewis</b></a></td>
      <td><a href="#Summers"
	     style="color:black"><b>Summers</b></a></td>
    </tr>

    <tr>
      <td>11:30</td>
      <td><a href="#Kirtland"
	     style="color:black"><b>Kirtland</b></a></td>
      <td><a href="#Kappe" style="color:black"><b>Kappe</b></a></td>
    </tr>

    <tr>
      <td>12:00</td>
      <td><a href="#Klepadlo"
	     style="color:black"><b>Klepadlo</b></a></td>
      <td class="break">Closing Remarks</td>
    </tr>
    

    <tr>
      <td>12:20</td>
<td class="break">Conference photograph</td>
</tr>

    <tr>
      <td>12:30</td>
    <td class="invited" rowspan="3">Lunch Break</td>
</tr>
    <tr><td>1:00</td>
    </tr>
    <tr><td>1:30</td></tr>
    
    <tr>
      <td>2:00</td>
      <td><a href="#Foguel" style="color:black"><b>Foguel</b></a></td>
    </tr>
    
    <tr><td>2:30</td>
      <td><a href="#Russell"
	     style="color:black"><b>Russell</b></a></td>
    </tr>

    <tr>
      <td>3:00</td>
      <td><a href="#Beike" style="color:black"><b>Beike</b></a></td>
    </tr>

    <tr>
      <td>3:30</td>
      <td style="break">Coffee Break</td>
    </tr>

    <tr>
      <td>4:00</td>
      <td><a href="#Feldman"
	     style="color:black"><b>Feldman</b></a></td>
    </tr>

    <tr>
      <td>4:30</td>
      <td><a href="#Martin" style="color:black"><b>Martin</b></a></td>
    </tr>

    <tr>
      <td>5:00</td>
      <td><a href="#Zaremsky"
	     style="color:black"><b>Zaremsky</b></a></td>
    </tr>
</tbody></table></div>
</section>

<section class="abstracts">
  <h2>Abstracts</h2>
  <div class="talk" id="Guzman">
    <p class="talk-title toggle"><b>The isomorphism theorems for
      pointed <i>g</i>-digroups</b></p>
    <p>Fernando Guzman (Binghamton University) Saturday 9:00 am</p>
    <p class="talk-abstract hidden">
Digroups, and generalized digroups, <i>g</i>-digroups for short,
have been considered as a generalization of continuous groups whose
tangent space is a Leibniz algebra. This structure has been seen as a
generalization of groups, therefore, efforts have been done to study
properties and results that come from group theory, to explore if they
hold in this new setting.  A pointed <i>g</i>-digroup is
a <i>g</i>-digroup with a 
distinguished bar-unit.<br><br>
In this talk, we'll discuss the isomorphism theorems for pointed
<i>g</i>-digroups, and show that the results for groups do extend to pointed
<i>g</i>-digroups.  Most of them also hold for <i>g</i>-digroups.
This is joint 
work with Olga Patricia Salazar-Diaz.</p>
</div>

  <div class="talk" id="Donoven">
    <p class="talk-title toggle"><b>Generation of simple vigorous
	groups of homeomorphisms</b></p>
    <p>Casey Donoven (Montana State University Northern) Saturday 9:30 am</p>
    <p class="talk-abstract hidden">
      A group of homeomorphisms <i>G</i> of Cantor space is vigorous if for any
      clopen subsets <i>B,C</i>&sub;<i>A</i> of Cantor space,
there exists a
&gamma;&isin;<i>G</i>
such that <i>B</i>&gamma;&sube;<i>C</i>.  Bleak, Hyde, and Elliot
(2024) proved that every finitely generated simple vigorous group is
2-generated.  In this talk, I will present recent findings extending
these results.  For example, if <i>G</i> is a f.g. simple vigorous group,
then  (i) <i>G</i> is generated by 3 involutions;
(ii) <i>G</i> is generated by an element of order <i>m</i> and <i>n</i> for all
<i>m</i>&ge;2 and <i>n</i>&ge;3, (iii)
<i>G</i> has a minimal generating set of size <i>k</i> for
all <i>k</i>&ge; 2, and (iv)
every nontrivial element of <i>G</i> is contained in a generating pair.
This is joint work with Collin Bleak, Scott Harper, and James
Hyde.</p>
    </div>

  <div class="talk" id="Tran">
    <p class="talk-title toggle"><b>Algebraic Combinatorics meets
	Probability Theory: Vines and MAT-labeled graphs</b></p>
    <p>Tan Tran (Binghamton University) Saturday 10:00 am</p>
    <p class="talk-abstract hidden">
This talk explores the connection between two concepts from distinct
areas of mathematics. The first concept, a vine, is a graphical model
used to represent dependent random variables. Initially introduced by
Joe (1994) and later formalized by Cooke (1997), vines have become an
active research area with applications in probability theory and
uncertainty analysis. The second concept, MAT-freeness, is a
combinatorial property in the theory of freeness of the logarithmic
derivation module of hyperplane arrangements. First studied by
Abe-Barakat-Cuntz-Hoge-Terao (2016) and further developed by
Cuntz-Muecksch (2020), MAT-freeness has been a topic of increasing
interest. In particular, for graphic arrangements, Tsujie and I
recently demonstrated that MAT-freeness is completely characterized by
the existence of certain edge-labeled graphs, known as MAT-labeled
graphs. I will show that there is a fascinating equivalence between
the categories of locally regular vines and MAT-labeled
graphs. Notably, this leads to an equivalence between the categories
of regular vines and MAT-labeled complete graphs. This work is joint
with H.M. Tran (Hanoi) and S. Tsujie (Hokkaido). </p>
  </div>

  <div class="talk" id="Lewis">
    <p class="talk-title toggle"><b>Self-normalizing subgroups</b></p>
    <p>Mark Lewis (Kent State University) Saturday 11:00 am</p>
    <p class="talk-abstract hidden">
We consider groups with few conjugacy classes of self--normalizing
subgroups. 
    </p>
  </div>

  <div class="talk" id="Kirtland">
    <p class="talk-title toggle"><b>2-covering numbers of finite
	groups</b></p>
    <p>Joe Kirtland (Marist University) Saturday 11:30 am</p>
    <p class="talk-abstract hidden">
A set of proper subgroups is a covering for a group <i>G</i> if its union
is the whole group. The minimal number of subgroups needed to cover
<i>G</i> is called the covering number of <i>G</i> and is denoted by
&sigma;<i>(G)</i>.   A study of coverings of the Paige loop motivates the
concept of a 2-covering for a group <i>G</i>, which is a set of proper
subgroups of <i>G</i> such that every pair of elements of <i>G</i> are
contained 
in at least one subgroup in the set.  The minimal number of subgroups
needed to 2-cover a group <i>G</i> is called the 2-covering number and
denoted by &sigma;<sub>2</sub><i>(G)</i>. Properties of 2-covering
numbers will be 
presented with the 2-covering number determined for finite nilpotent
groups, finite almost simple groups, and particular classes of finite
solvable groups. 
</p></div>

  <div class="talk" id="Klepadlo">
    <p class="talk-title toggle"><b>Coverings of dihedral and
	permutation groups using centralizers</b></a>
<p>Matthew Klepadlo (Adelphi University) Saturday 12:00</p>
<p class="talk-abstract hidden">
A group is said to be covered if there exists proper subgroups such
that their union is the same as the whole group. This paper will go
into how we use centralizer subgroups to come up with coverings of
smaller dihedral and permutation groups and obtain the "covering
number" and "centralizer-covering number." We will also be
highlighting a few notable theorems regarding coverings and use them
to our advantage to finding said "covering/centralizer-covering
number." The history of coverings and the interest mathematicians have
in them will also be explored. </p></div>

<div class="talk" id="Foguel">
  <p  class="talk-title toggle"><b>Finite groups in which every
      subgroup of order divisible by <i>p</i> is normal</b></p>
  <p>Tuval Foguel (Adelphi University) 2:00 pm</p>
  <p class="talk-abstract hidden">
In this talk, I’ll introduce two generalizations of Dedekind groups,
called <i>PN</i>-groups and <i>PNQ</i>-groups. In <i>PN</i>-groups,
every subgroup 
whose order is divisible by a fixed prime <i>p</i> is normal, while in
<i>PNQ</i>-groups, such subgroups are permutable. We’ll begin by showing
that these groups must be either <i>p</i>'-groups or
supersolvable. From there, I’ll walk through a classification of both
<i>PN</i>- and <i>PNQ</i>-groups. I’ll end with a brief discussion of minimal
non-<i>PN</i>-groups and some questions that remain open. </p></div>

<div class="talk" id="Russell">
  <p class="talk-title toggle"><b>Extended Springer fibers
      overview</b></p>
  <p>Amber Russell (Butler University) Saturday 2:30 pm</p>
  <p class="talk-abstract hidden">
The Springer Correspondence associates to each irreducible
representation of the Weyl group for a reductive Lie algebra a
nilpotent orbit for the Lie algebra and an irreducible representation
of the fundamental group of the Lie algebra.  This result was due to
T.A. Springer in the 1970s and is still providing fertile grounds of
innovation today.  The key tools in this work were the Springer
resolution, a resolution of singularities for the nilpotent cone of
the Lie algebra, and also careful study of the resulting Springer
fibers. In the 1980s, George Lusztig expanded this to a bijection
where all possible pairs of nilpotent orbits and irreducible
representations of the fundamental group appear and the Weyl group is
replaced by a class of new relative Weyl groups.  This is Lusztig's
Generalized Springer Correspondence.<br><br>
Over the past several years, I have collaborated will William Graham
and Martha Precup on a related project, spanning multiple publications
with a new one currently being prepared.  In particular, we have
studied Extended Springer Fibers and connected them to Lusztig's
Generalized Springer Correspondence in all classical types and
relevant exceptional types.  The goal of this presentation will be an
overview of these results. 
</p></div>

<div class="talk" id="Beike">
  <p class="talk-title toggle"><b><i>p</i>-Groups with derived length
      three and three character degrees</b></p>
<p>Nic Beike (Kent State University) Saturday 3:00 pm</p>
  <p class="talk-abstract hidden">
We will construct examples of <i>p</i>-groups with derived length 3 and 3
character degrees. We will focus on groups of order <i>p</i><sup>6</sup>. 
</p></div>

<div class="talk" id="Feldman">
  <p class="talk-title toggle"><b>Another dual to Schunck
      classes</b></p>
  <p>Arnold Feldman (Franklin and Marshall College) Saturday 4:00
    pm</p>
  <p class="talk-abstract hidden">
The duality of Fitting classes and formations has been widely studied,
and Fitting classes and Schunck classes can also be considered
to be dual in some sense. However, the definitions of Fitting
and Schunck classes are not literally dual the way that those of
Fitting classes and formations are.  Here we identify a dual to
Schunck classes of finite groups, which we call SchunckD
classes, based on the standard definition of a Schunck class.
We investigate properties and examples of Schunck classes and
see how they differ from Fitting classes.  The arguments are
relatively elementary, and the topic could lend itself to
exploration by advanced undergraduates.
</p></div>

<div class="talk" id="Martin">
  <p class="talk-title toggle"><b>Groups with a fixed character
      degree</b></p>
  <p>Brandon Martin (Kent State University) Saturday 4:30 pm</p>
  <p class="talk-abstract hidden">
Let <i>x=d<sub>1</sub>&hellip;d<sub>
      m </sub>p<sub>1</sub><sup>a<sub>1</sub></sup> &hellip; p<sub>n</sub><sup>a<sub>n</sub></sup></i>
where the <i>d<sub>j</sub></i>'s and <i>p<sub>i</sub></i>'s are
      distinct primes, and <i>a<sub>i</sub></i>&isin;<b>N</b>
for all <i>i</i>. Let 
<i>d=d<sub>1</sub>&hellip;d<sub>m</sub></i>.  We've previously shown there
exists a solvable 
group <i>G</i>, of order <i>x</i>, with <i>d</i>&isin;cd<i>(G)</i> if
and only if there 
is a sequence of congruences between the <i>p<sub>i</sub></i>'s and <i>d<sub>j</sub></i>'s where the
product of the moduli of these congruences is precisely <i>d</i>.  Now, we
let relax the square-free condition on <i>d</i> and consider when an
analogous result holds when <i>x=d<sup>m</sup>p<sup>n</sup></i>. 
</p></div>

<div class="talk" id="Zaremsky">
  <p class="talk-title toggle"><b>Some difficult simple groups</b></p>
  <p>Matt Zaremsky (University at Albany) Saturday 5:00 pm</p>
  <p class="talk-abstract hidden">
Finite simple groups are famously classified, but infinite simple
groups remain extremely mysterious in general. In particular, a famous
conjecture of Boone and Higman predicts that every finitely generated
group with solvable word problem embeds in a finitely presented simple
group, so finitely presented simple groups are conjecturally
ubiquitous, but actual examples are hard to come by. In this talk I
will survey some of the bizarre and interesting (infinite) simple
groups that arise, and mention some recent results, with a focus on a
family of simple groups called twisted Brin-Thompson
groups. </p></div>

<div class="talk" id="Riedl">
  <p class="talk-title toggle"><b>Images of iterated commutators under
      group automorphisms</b></a>
<p>Jeffrey Riedl (University of Akron) Sunday 8:30 am</p>
<p class="talk-abstract hidden">
Let <i>x</i>, <i>y</i> be elements of a group <i>G</i>. For each
integer <i>m</i>&ge;0 let 
<i>d<sub>m</sub></i> denote the <i>m</i>th iterated commutator of <i>x</i>
  by <i>y</i>. Thus
<i>d<sub>0</sub>=x</i>, <i>d<sub>1</sub>=[x,y]</i>, <i>d<sub>2</sub>=[[x,y],y]</i>,
and so on. Let <b>D</b><i>=
={d<sub>0</sub>,d<sub>1</sub>, d<sub>2</sub>,&hellip;}</i> and suppose
all the elements of <b>D</b>
commute with each other. Let &sigma; be an automorphism of <i>G</i> such
that <i>x<sup>&sigma;</sup>=x</i>
and <i>y<sup>&sigma;</sup>=y<sup>u</sup></i> for some positive integer 
<i>u</i>. We establish a formula that expresses the image under &sigma; of
an arbitrary element of <b>D</b> as a product of elements of <b>D</b>.
<br><br>
We mention the application that motivated the establishment of this
formula. Let <i>p</i> be a prime and let <i>n</i>&ge;2. The group
U(<i>p<sup>n</sub></i>) of
units modulo <i>p<sup>n</sup></i> acts naturally via automorphisms on
the regular 
wreath product group <i>W=Z<sub>p<sup>2</sup></sub></i> wr
  <i>Z<sub>p</sup>n</sup></sub></i>, and hence acts on the set
<b>N</b> consisting of all the normal subgroups of <i>W</i> that are
contained in the base group of <i>W</i>. The formula enables the
straightforward computation of the image <b>N</b><sup>&sigma;</sup> of
an arbitrary 
<i>N</i>&isin;<b>N</b> for an arbitrary automorphism &sigma;&isin;
U(<i>p<sup>n</sup></i>).  
</p></div>

<div class="talk" id="McCulloch">
  <p class="talk-title toggle"><b>Groups with dense Chermak-Delgado
      subgroups</b></p>
  <p>Ryan McCulloch (Binghamton University) Sunday 9:00 am</p>
  <p class="talk-abstract hidden">
Let <b><i>X</i></b> be a property pertaining to subgroups of a group. We
say that a group 
<i>G</i> has dense <b><i>X</i></b>-subgroups if for each pair <i>(H, K)</i> of
subgroups of <i>G</i> such that <i>H</i> &lt;<i>K</i> and <i>H</i> is
not maximal in <i>K</i>, 
there exists an <b><i>X</i></b>-subgroup <i>X</i> of <i>G</i> such
that <i>H &lt; X &lt;  
K</i>.  In this talk we consider groups with Chermak--Delgado dense
subgroups and, more generally, with centralizer dense subgroups.  This
includes joint work with Marius Tarnauceanu.    </p></div>

<div class="talk" id="Cinarci">
  <p class="talk-title toggle"><b>Some results on derived length and
      character degrees</b></p>
  <p>Burcu &Ccedil;&imath;narc&imath; (Texas State University)
    Sunday 9:30 am</p>
  <p class="talk-abstract hidden">
The character degrees of a finite group provide some important
information about the structure of the group. A famous problem on the
character degrees of a finite solvable group <i>G</i> is known as the
Taketa problem and Isaacs-Seitz conjecture. This problem states that
the inequality <i>dl(G) &le; |cd(G)|</i> holds for a finite solvable group
<i>G</i>, where <i>dl(G)</i> is the derived length of <i>G</i> and
<i>|cd(G)|</i> is the
cardinality of the set of all irreducible character degrees of
<i>G</i>. Although this conjecture is still open, many research articles
have been published on this inequality. In this talk, we show that the
Taketa inequality holds for <i>G</i> under some sufficient conditions.
</p></div>

<div class="talk" id="Zarrin">
  <p class="talk-title toggle"><b>On the noncommuting set in infinite
      groups</b></p>
  <p>Mohammad Zarrin (Texas State University) Sunday 10:30 am</p>
  <p class="talk-abstract hidden">
Let <i>G</i> be a non-abelian group. A subset <i>T</i> of a
group <i>G</i> is a set of 
pairwise noncommuting elements if <i>xy&neq; yx</i> for any two distinct
elements <i>x</i> and <i>y</i> in <i>T</i>.<br><br>
If <i>|T| &ge; |R|</i> for any other set of pairwise noncommuting elements
<i>R</i> in <i>G</i>, then <i>T</i> is called a maximal subset of pairwise
noncommuting elements and the cardinality of such a subset (if it
exists) is denoted by <i>w(G)</i>. In this talk, among other things, we
show that, for each positive integer <i>m</i>, there are only finitely many
groups <i>G</i>, up to isoclinism, with <i>w(G) = m</i>, and we obtain similar
results for groups with exactly <i>m</i> centralizers.<br><br>
Also, we try to find the influence of the function <i>w(G)</i> on the
structure of groups. </p></div>

<div class="talk" id="Summers">
  <p class="talk-title toggle"><b>On the number of disconnected
      character degree graphs satisfying P&aacute;lfy's inequality</b></p>
  <p>Andrew Summers (Kent State University) Sunday 11:00 am</p>
  <p class="talk-abstract hidden">
Let <i>G</i> be a finite solvable group with disconnected character degree
graph &Delta;(<i>G</i>). Under these conditions, it follows from a result of
P&aacute;lfy that &Delta;(<i>G</i>) consists of two connected
components. Another 
result of P&aacute;lfy's gives an inequality relating the sizes of these two
connected components. In this talk, some background on character
degree graphs and P&aacute;lfy's results will be presented. The number of
possible component size pairs that satisfy Pálfy's inequality will be
calculated. Additionally, for a fixed positive integer <i>n</i>, the number
of distinct graph orders for which exactly <i>n</i> component size pairs
satisfy P&aacute;lfy's inequality is shown. 
</p></div>

<div class="talk" id="Kappe">
  <p class="talk-title toggle"><b>Element centralizers in a group
      centralizer lattice and centralizer-like subgroups</b></p>
  <p>Luise-Charlotte Kappe (Binghamton University) Sunday 11:30 am</p>
  <p class="talk-abstract hidden">
We note some properties of the centralizer map and recall the
centralizer lattice of a group. Since the element centralizers
generate all the other centralizers, we consider how the element
centralizers sit in the lattice. We generalize this by considering the
so-called centralizer-like subgroups of a group associated with a
2-letter word <i>w(u,v)</i>. These are four subgroups defined by an
operator that takes as input a subgroup <i>H</i> and returns the subgroup
of group elements <i>x</i> such
that <i>w(xg,h)=w(g,h)</i>, <i>w(gx,h)=w(g,h)</i>, 
<i>w(h,xg)=w(h,g)</i>, and <i>w(h,gx) = w(h,g)</i> respectively, for
all <i>g</i>&isin;<i>G</i> and <i>h</i>&isin;<i>H</i>. We investigate
for which words these centralizer-like 
subgroups also generate a lattice that is a centralizer-like lattice.
<br><br>
This is joint work with Wil Cocke, Mark Lewis, and Ryan McCulloch.
</p></div>		   

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