Group Theory Abstracts

On a Question of Glasby, Praeger, and Xia

MICHAEL JAMES BARRY, Allegheny College

Recently, Glasby, Praeger, and Xia asked for necessary and sufficient conditions for the `Jordan Partition' $\lambda(r,s,p)$ to be standard. We give such conditions when $p$ is an odd prime.

ALLEGHENY COLLEGE
MEADVILLE, PA
mbarry at allegheny.edu

Groups with all order subsets dividing the order of the group

BRET JORDAN BENESH, College of Saint Benedict and Saint John's University

Consider the symmetric group $S_3$ acting on three letters. It has one element of order $1$, three elements of order $2$, and two elements of order $3$. Notice that for each order, the number of elements of that order (one, three, and two, respectively) divide the order of the group, which is $3!6$.

Groups with this property are called``perfect order subset groups,†or POS groups. Finch and Jones have already done a lot of work on abelian POS groups. This talk--based on work done with Shadow Zhao--will focus on nonabelian POS groups. In particular, we classify all nonabelian POS groups of the form $\mathbb{Z}_m \rtimes \mathbb{Z}_n$, where the action is inversion.

COLLEGE OF SAINT BENEDICT AND SAINT JOHN'S UNIVERSITY
SAINT JOSEPH, MN
bbenesh at csbsju.edu

On Chermak-Delgado Simple $p$-Groups

JOSEPH BRENNAN, Binghamton University SUNY

The Chermak-Delgado measure for a group $G$ is $\mathfrak{M}_G = \max\{\vert H\vert\vert C_G(H)\vert\,:\, H\leq G\}$, while the Chermak-Delgado lattice for a group $G$ is the set $\CD (G) =\{ H\leq G\,:\, \vert H\vert\vert C_G(H)\vert \mathfrak{M}_G \}$ ordered by inclusion.

A group G is CD-simple if $CD(G)\{ G, Z(G) \}$; it should be noted that abelian, simple, and the symmetric groups are CD-simple. The groups of interest for this talk are non-abelian finite $p$-groups and we begin with a few structure results for such groups which are CD-simple. An infinite family of CD-simple $p$-groups, elements of which may have an arbitrary odd nilpotence class, will be presented. Using the structure results the first non-abelian CD-simple p-groups will be determined (first as in minimal $n$ with respect to some odd prime $p$ and order $p^n$).

This is joint work with Lijian An and HaiPeng Qu from Shaanxi Normal University, China.

BINGHAMTON UNIVERSITY SUNY
BINGHAMTON, NY
jbrennan at binghamton.edu

Dualizable Groups

ERAN CROCKETT, Binghamton University

Many are familiar with the Pontryagin Duality between abelian groups and compact topological abelian groups. Similar results hold for non-abelian groups if we allow the use of more general topological structures as opposed to topological groups. This talk is a survey of the results of Davey, Quackenbush, Szabó, and Nickodemus.

BINGHAMTON UNIVERSITY
BINGHAMTON,
crockett at math.binghamton.edu

Simple Right Conjugacy Closed Loops and Hall Planes

MARK BENJAMIN GREER, University of North Alabama

A loop $Q$ is a right conjugacy closed loop (or RCC loop) if $R_{Q}$ is closed under conjugation. In this talk, we give the first general construction of a large class of nonassociative, finite simple RCC loops. Our construction by no means accounts for all such loops, thus a full classification of finite simple RCC loops is still elusive. It turns out that our construction is not new, but is easily seen to be equivalent to a construction by Hall for non-Desarguesian planes, called Hall planes. Though Hall planes of the same order are isomorphic, the same is not true for the associated RCC loops. Hence, we will discuss the classification of such loops.

UNIVERSITY OF NORTH ALABAMA
FLORENCE, ALABAMA
mgreer at una.edu

Variations on a Theme of I.D. Macdonald

LUISE-CHARLOTTE KAPPE, Binghamton University

In a $1963$ paper I.D. Macdonald gave an example of a group in which the cyclic commutator subgroup is not generated by a commutator and he gives sufficient conditions on the group G such that its cyclic commutator subgroup is generated by a commutator.

The question arises, what is the situation for other words in case the associated word subgroup is cyclic, in particular the word $x^n$, $n$ a positive integer. For $n$ a positive integer, we establish sufficient conditions such that $G^n = \left\langle g^n: g\in G\right\rangle$ is generated by an $n^{th}$ power in case $G^n$ is cyclic and give examples of groups G, were $G^m$ is cyclic but not generated by the $n^{th}$ power of an element.

Joint work with: Joseph Brennan of Binghamton University Gabriela Mendoza of Riverside Community College

BINGHAMTON UNIVERSITY
BINGHAMTON, NEW YORK
menger at math.binghamton.edu

Random Groups

ANDREW J KELLEY, Binghamton University

What does a typical group look like? What properties does one usually have? Misha Gromov developed several models of random presentations to answer this question for finitely presented groups. One or two different models will be described as well as some results.

BINGHAMTON UNIVERSITY
VESTAL, NY
akelley3 at binghamton.edu

Closure Operators on a Subgroup Lattice: Preliminary Report

MARTHA LEE H. KILPACK, SUNY Oneonta

Starting with a lattice which is isomorphic to a subgroup lattice, $Sub(G)$, we take all the closure operators on that lattice and create a new lattice, the lattice of closure operators of $Sub(G)$, $c.o.(Sub(G))$. We will show for $G Z_{pq}$ where $p\neq q$ and both are prime, $c.o.(Sub(G))$ is not isomorphic to a subgroup lattice. We will look at other groups that have similar results leading us to a conjecture about $c.o.(Sub(G))$ for finite groups.

SUNY ONEONTA
ONEONTA, NY
martha.kilpack at oneonta.edu

On Two Classes of Finite Inseparable $p$-Groups

JOSEPH KIRTLAND, Marist College

A finite group is inseparable it does not split over any proper nontrivial normal subgroup; that is, if it has no nontrivial semidirect product decompositions. This talk investigates two classes of finite inseparable $p$-groups and, for $p \geq 3$, establishes a necessary and sufficient condition for inseparability.

MARIST COLLEGE
POUGHKEEPSIE, NEW YORK
joe.kirtland at marist.edu

An application of the Artin-Hasse exponential to finite algebra groups

DARCI L. KRACHT, Kent State University

An algebra group is a group of the form $G 1+J$, where $J$ is a finite- dimensional, nilpotent, associative algebra over a field of prime characteristic $p$. Important classes of subgroups of $G$ are defined in terms of the ordinary exponential series, but these make sense only if $J^p 0$. We will define analogs of these subgroups that make sense in the general setting using the Artin-Hasse exponential series and use them to answer a question about normalizers.

KENT STATE UNIVERSITY
KENT, OH
darci at math.kent.edu

A proof that a 4-generated $p$-group of class at most two and prime

exponent is either cyclic, extra-special, or capable, using algebraic geometry.

ARTURO MAGIDIN, University of Louisiana at Lafayette

A group $G$ is capable if $G\cong K/Z(K)$ for some $K$. It has long been known that nontrivial cyclic groups and extra-special $p$-groups of order greater than $p^3$ cannot be capable. The capability of groups of class 2 and prime exponent can be characterized in terms of certain subspaces and linear transformations between vector spaces over $\mathbb{F}_p$, and this set-up opens the door to other tools, in particular geometric tools, to enter the picture. In particular, we will show an argument using algebraic geometry to show that if $G$ is of class two and prime exponent, and $\vert G^{\rm ab}\vert\leq p^4$, then the nontrivial cyclic group and the extra-special group of order $p^5$ and exponent $p$ are in fact the only exceptions to capability. That is, such a group $G$ is either non-trivial cyclic, extra-special of order $p^5$, or capable. The proof includes joint work with David McKinnon (University of Waterloo).

UNIVERSITY OF LOUISIANA AT LAFAYETTE
LAFAYETTE, LOUISIANA
magidin at member.ams.org

The Chermak-Delgado Lattice of Finite Groups

RYAN J. MCCULLOCH, Binghamton University

We will present new results on the Chermak-Delgado lattice of finite groups. An emphasis will be on non p-groups.

BINGHAMTON UNIVERSITY
VESTAL, NY
mcculloch at math.binghamton.edu

On the character degree ratio of finite groups

HUNG NGUYEN, The University of Akron

For a finite nonabelian group $G$ let $rat(G)$ be the largest ratio of degrees of two nonlinear irreducible characters of $G$. The character degree ratio was first introduced and studied by I.M. Isaacs in connection with character kernels in finite groups. For instance, Isaacs proved that the derived length and the Fitting height of a solvable group $G$ are bounded above respectively by $3+4\log_2(rat(G))$ and $3+2\log_2(rat(G))$. This result indicates that the derived length and the Fitting height of a solvable group are controlled by its character degree ratio.

We will show that composition factors of an arbitrary finite group $G$ are somehow also controlled by $rat(G)$. Specifically, if $S$ different from the simple linear groups $PSL_2(q)$ is a nonabelian composition factor of $G$, then the order of $S$ and the number of composition factors of $G$ isomorphic to $S$ are both bounded in terms of $rat(G)$. Furthermore, when the groups $PSL_2(q)$ are not composition factors of $G$, we prove that $\vert G:O_\infty(G)\vert\leq rat(G)^{21}$ where $O_\infty(G)$ denotes the solvable radical of $G$.

THE UNIVERSITY OF AKRON
AKRON, OHIO
hn10 at uakron.edu

Finitely Constrained Groups of Maximal Hausdorff Dimension

ANDREW DANIEL PENLAND, Texas A&M University

A group of binary tree automorphisms is called finitely constrained if the action of each element on the tree can be described in terms of a single finite group. By viewing the group of all rooted binary tree automorphisms as a metric space, we can associate a Hausdorff dimension to the finitely constrained subgroups. In this talk, we will show that finitely constrained groups of the largest possible Hausdorff dimension are not topologically finitely generated.

This is a joint work with Zoran Šunic.

TEXAS A M UNIVERSITY
BRYAN, TX
apenland at math.tamu.edu

On the covering numbers of small symmetric and sporadic groups

DANIELA BORISLAVOVA POPOVA, Florida Atlantic University

ON THE COVERING NUMBER OF SMALL SYMMETRIC GROUPS AND SOME SPORADIC SIMPLE Speaker: Daniela Nikolova-Popova Department of Mathematical Sciences, Florida Atlantic University, USA E:mail: dpopova at fau.edu Joint work with Luise-Charlotte Kappe, and Eric Swartz

We say that a group G has a finite covering if G is a set theoretical union of finitely many proper subgroups. The minimal number of subgroups needed for such a covering is called the covering number of G denoted by Ï­(G). Let Sn be the symmetric group on n letters. For odd n Maroti determined Ï­(Sn) with the exception of n 9, and gave estimates for n even showing that Ï­(Sn) ≀ 2n-2. Using GAP calculations, as well as incidence matrices and linear programming, we show that Ï­(S8) 64, Ï­(S10) 221, Ï­(S12) 761. We also show that Maroti ’s result for odd n holds without exception proving that Ï­(S9)We establish in addition that the Mathieu group M12 has covering number 208, and improve the estimate for the Janko group J1 given by P.E. Holmes. References: [1] A. Maroti, Covering the symmetric groups with proper subgroups, J. Comb. Theory Ser.A 110 (2005), 97-111. [2] P.E. Holmes, Subgroup coverings of some sporadic groups, J. Comb. Theory Ser. A 113(2006), 1204-1213.

FLORIDA ATLANTIC UNIVERSITY
BOCA RATON, FL
dpopova at fau.edu

Moufang semidirect products of loops with groups and inverse property extensions

LEE STEPHEN RANEY, University of North Alabama

We investigate loops which can be written as the semidirect product of a loop and a group, and we provide a necessary and sufficient condition for such a loop to be Moufang. We also examine a class of loop extensions which arise as a result of a finite cyclic group acting as a group of semiautomorphisms on an inverse property loop. In particular, we consider closure properties these from an external point of view. This is joint work with Mark Greer.

UNIVERSITY OF NORTH ALABAMA
FLORENCE, ALABAMA
lraney at una.edu

Residual finiteness of multiple ascending HNN extensions

SLOBODAN TANUSEVSKI, Binghamton University

I will describe few results on the residual finiteness of multiple ascending HNN extensions. This is part of a joint work with Xiaolei Wu.

BINGHAMTON UNIVERSITY
BINGHAMTON,
tanusevski at math.binghamton.edu

Large abelian subgroups of finite groups

ALEXANDRE TURULL, University of Florida

A celebrated theorem of C. Jordan states the following. For any $d$ there exists a constant $C$ such that any finite subgroup $G$ of $\GL (d,\mathbf{C})$ has an abelian subgroup $A\subseteq G$ of index at most $C$, and $A$ can be generated by at most $d$ elements. Analogues of this theorem have been proposed where the new statements concern finite subgroups of important infinite groups. We discuss some of these conjectures and results. We discuss joint work with Ignasi Mundet i Riera where the proof of these conjectures is reduced to proving some much more limited statements. These joint results were used by Mundet to prove a celebrated conjecture of Étienne Ghys for manifolds without odd cohomology, as well as for other manifolds. They were also used by B. Zimmermann to prove that the conclusion of Jordan's theorem also holds for yet other smooth manifolds.

UNIVERSITY OF FLORIDA
GAINESVILLE, FLORIDA
turull at ufl.edu

Magic and Orthogonal Cayley-Sudoku Tables

MICHAEL WARD, Western Oregon University

A Cayley-Sudoku Table (C-S Table) is the Cayley table of a finite group arranged (unconventionally) so that the body of the Cayley table is divided into blocks containing each group element exactly once, as in a sudoku puzzle. In this talk we consider Magic C-S Tables in which each block of the C-S Table is a magic square. That is, the (group) sum of the entries in each row, column, and diagonal of each block equals the same fixed group element. We also discuss orthogonal C-S tables, in the sense of Latin square orthogonality. Open questions suitable for undergraduate investigation are included. This is joint work with Rosanna Mersereau (WOU `13), The Ohio State University.

WESTERN OREGON UNIVERSITY
MONMOUTH, OREGON
wardm at wou.edu

Quasiantichains as Chermak-Delgado lattices of p-Groups

ELIZABETH WILCOX, Oswego State University

The Chermak-Delgado lattice of a finite group $G$ is a sublattice of the subgroup lattice of $G$. The Chermak-Delgado lattice, denoted $\mathcal{C}\mathcal{D} (G)$, has several interesting properties that make it a fascinating idea to study. In this talk, we'll let $p$ be a prime and consider only $p$-groups, $P$, where $\mathcal{C}\mathcal{D} (P)$ is known to be a quasiantichain of width at least 3. Our goal will be to answer, as best as currently able, the following questions: What can be said about the structure of $P$? What can be said about the subgroups in $\mathcal{C}\mathcal{D} (P)$? What can be said about the width, or number of atoms, in $\mathcal{C}\mathcal{D} (P)$?

OSWEGO STATE UNIVERSITY
OSWEGO, NY
elizabeth.wilcox at oswego.edu

Prime power divisors of character degrees

THOMAS WOLF, Ohio University

If $p^2$ does not divide $X(1)$ for all irreducible characters $X$ of a solvable group $G$, then a Sylow-p-subgroup of $G/F(G)$ has order at most $p^2$. If we drop solvability and assume $p=2$, then a Sylow-2-subgroup of $G/F(G)$ has order at most $8$. This is joint work with M. Lewis and G. Navarro to appear in J. Algebra.

OHIO UNIVERSITY
ATHENS, OHIO
wolf at ohiou.edu

Normal subgroups of braided Thompson groups

MATT ZAREMSKY, Binghamton University

The braided Thompson groups were developed by Matt Brin and, independently, Patrick Dehornoy. The braided Thompson group $V_{br}$ contains Thompson's group F and also copies of every braid group. Despite being so vast as to contain every braid group, $V_{br}$ has nice finiteness properties, e.g., it is finitely presented. I will discuss recent work characterizing the normal subgroups of these groups, including a result that every proper quotient of $V_{br}$ descends to the natural quotient onto Thompson's group $V$.

BINGHAMTON UNIVERSITY
BINGHAMTON, NY
zaremsky at math.binghamton.edu