Searcharxiv⌕ Search

arXiv subjects

Eric Swartz

Publications and source records attributed to Eric Swartz.

At least 19 recordsLinked to original sources
arXiv2026

Circular sorting in the alternating group

The symmetric group $S_n$ is generated by transpositions, and problems of sorting permutations using transpositions are well studied. In recent work, Adin, Alon, and Roichman studied the related problem of sorting $n$ points on a circle, and gave a formula for the maximum number of adjacent swaps required. This is equivalent to the number of adjacent transpositions required to transform any permutation into a power of the cyclic permutation $(1,2,\ldots, n)$. The focus of this work is an analogous question in the alternating group $A_n$, which is generated by $3$-cycles. That is, using 3-cycles instead of transpositions, what is the maximum number of steps required to transform an even permutation into a power of $(1,2,\ldots, n)$ in the alternating group? We determine this number exactly for even $n$ and $n \equiv 1 \pmod{4}$. For $n \equiv 3 \pmod{4}$, we show that the sorting number can take one of two possible values and give explicit constructions demonstrating that the larger value occurs infinitely often.

math.CO↗
arXiv2025

Covering rings by proper ideals

A cover by left ideals of an associative (not necessarily commutative or unital) ring $R$ is a collection of proper left ideals whose set-theoretic union equals $R$. If such a cover exists, then $\eta_\ell(R)$ is the cardinality of a minimal cover, and $R$ is $\eta_\ell$-elementary if $\eta_\ell(R)<\eta_\ell(R/I)$ for every nonzero two-sided ideal $I$ of $R$. We classify all $\eta_\ell$-elementary rings, and determine their covering numbers. Covers by right or two-sided ideals are also studied. This completely characterizes rings admitting finite covers by ideals. Our results generalize to finite covers of modules by submodules, and we determine all possible covering numbers.

math.RA↗
arXiv2025

Enumerating Vector Parking Functions and their Outcomes Based on Specified Lucky Cars

In a parking function, a car is considered lucky if it is able to park in its preferred spot. Extending work of Harris and Martinez, we enumerate outcomes of parking functions with a fixed set of lucky cars. We then consider a generalization of parking functions known as vector parking functions or $\boldsymbol{u}$-parking functions, in which a nonnegative integer capacity is given to each parking spot in the street. With certain restrictions on $\boldsymbol{u}$, we enumerate outcomes of $\boldsymbol{u}$-parking functions with a fixed set of lucky cars or with a fixed number of lucky cars. We also count outcomes according to which spots contain lucky cars, and give formulas for enumerating $\boldsymbol{u}$-parking functions themselves according to their set of lucky cars.

math.CO↗
arXiv2025

Character theoretic techniques for nonabelian partial difference sets

A $(v,k,\lambda, \mu)$-partial difference set (PDS) is a subset $D$ of size $k$ of a group $G$ of order $v$ such that every nonidentity element $g$ of $G$ can be expressed in either $\lambda$ or $\mu$ different ways as a product $xy^{-1}$, $x, y \in D$, depending on whether or not $g$ is in $D$. If $D$ is inverse closed and $1 \notin D$, then the Cayley graph ${\rm Cay}(G,D)$ is a $(v,k,\lambda, \mu)$-strongly regular graph (SRG). PDSs have been studied extensively over the years, especially in abelian groups, where techniques from character theory have proven to be particularly effective. Recently, there has been considerable interest in studying PDSs in nonabelian groups, and the purpose of this paper is develop character theoretic techniques that apply in the nonabelian setting. We prove that analogues of character theoretic results of Ott about generalized quadrangles of order $s$ also hold in the general PDS setting, and we are able to use these techniques to compute the intersection of a putative PDS with the conjugacy classes of the parent group in many instances. With these techniques, we are able to prove the nonexistence of PDSs in numerous instances and provide severe restrictions in cases when such PDSs may still exist. Furthermore, we are able to use these techniques constructively, computing several examples of PDSs in nonabelian groups not previously recognized in the literature, including an infinite family of genuinely nonabelian PDSs associated to the block-regular Steiner triple systems originally studied by Clapham and related infinite families of genuinely nonabelian PDSs associated to the block-regular Steiner $2$-designs first studied by Wilson.

math.CO↗
arXiv2024

Combinatorial transfer: a new method for constructing infinite families of nonabelian difference sets, partial difference sets, and relative difference sets

For nearly a century, mathematicians have been developing techniques for constructing abelian automorphism groups of combinatorial objects, and, conversely, constructing combinatorial objects from abelian groups. While abelian groups are a natural place to start, recent computational evidence strongly indicates that the vast majority of transitive automorphism groups of combinatorial objects are nonabelian. This observation is the guiding motivation for this paper. We propose a new method for constructing nonabelian automorphism groups of combinatorial objects, which could be called the \textit{combinatorial transfer method}, and we demonstrate its power by finding (1) the first infinite families of nonabelian Denniston partial difference sets (including nonabelian Denniston PDSs of odd order), (2) the first infinite family of Spence difference sets in groups with a Sylow 3-subgroup that is non-normal and not elementary abelian, (3) the first infinite families of McFarland difference sets in groups with a Sylow $p$-subgroup that is non-normal and is not elementary abelian, (4) new infinite families of partial difference sets in nonabelian $p$-groups with large exponent, (5) an infinite family of semiregular relative difference sets whose forbidden subgroup is nonabelian, and (6) a converse to Dillon's Dihedral Trick in the PDS setting. We hope this paper will lead to more techniques to explore this largely unexplored topic.

math.CO↗
arXiv2024

Exponential and weakly exponential subgroups of finite groups

Sabatini (2024) defined a subgroup $H$ of $G$ to be an exponential subgroup if $x^{|G:H|} \in H$ for all $x \in G$. Exponential subgroups are a generalization of normal (and subnormal) subgroups: all subnormal subgroups are exponential, but not conversely. Sabatini proved that all subgroups of a finite group $G$ are exponential if and only if $G$ is nilpotent. The purpose of this paper is to explore what the analogues of a simple group and a solvable group should be in relation to exponential subgroups. We say that an exponential subgroup $H$ of $G$ is exp-trivial if either $H = G$ or the exponent of $G$, ${\rm exp}(G)$, divides $|G:H|$, and we say that a group $G$ is exp-simple if all exponential subgroups of $G$ are exp-trivial. We classify finite exp-simple groups by proving $G$ is exp-simple if and only if ${\rm exp}(G) = {\rm exp}(G/N)$ for all proper normal subgroups $N$ of $G$, and we illustrate how the class of exp-simple groups differs from the class of simple groups. Furthermore, in an attempt to overcome the obstacle that prevents all subgroups of a generic solvable group from being exponential, we say that a subgroup $H$ of $G$ is weakly exponential if, for all $x \in G$, there exists $g \in G$ such that $x^{|G:H|} \in H^g$. If all subgroups of $G$ are weakly exponential, then $G$ is wexp-solvable. We prove that all solvable groups are wexp-solvable and almost all symmetric and alternating groups are not wexp-solvable. Finally, we completely classify the groups ${\rm PSL}(2,q)$ that are wexp-solvable. We show that if $\pi(n)$ denotes the number of primes less than $n$ and $w(n)$ denotes the number of primes $p$ less than $n$ such that ${\rm PSL}(2,p)$ is wexp-solvable, then $\lim_{n \to \infty} \frac{w(n)}{\pi(n)} = \frac{1}{4}.$

math.GR↗
arXiv2023

Nonabelian partial difference sets constructed using abelian techniques

A $(v,k,\lambda, \mu)$-partial difference set (PDS) is a subset $D$ of a group $G$ such that $|G| = v$, $|D| = k$, and every nonidentity element $x$ of $G$ can be written in either $\lambda$ or $\mu$ different ways as a product $gh^{-1}$, depending on whether or not $x$ is in $D$. Assuming the identity is not in $D$ and $D$ is inverse-closed, the corresponding Cayley graph ${\rm Cay}(G,D)$ will be strongly regular. Partial difference sets have been the subject of significant study, especially in abelian groups, but relatively little is known about PDSs in nonabelian groups. While many techniques useful for abelian groups fail to translate to a nonabelian setting, the purpose of this paper is to show that examples and constructions using abelian groups can be modified to generate several examples in nonabelian groups. In particular, in this paper we use such techniques to construct the first known examples of PDSs in nonabelian groups of order $q^{2m}$, where $q$ is a power of an odd prime $p$ and $m \ge 2$. The groups constructed can have exponent as small as $p$ or as large as $p^r$ in a group of order $p^{2r}$. Furthermore, we construct what we believe are the first known Paley-type PDSs in nonabelian groups and what we believe are the first examples of Paley-Hadamard difference sets in nonabelian groups, and, using analogues of product theorems for abelian groups, we obtain several examples of each. We conclude the paper with several possible future research directions.

math.CO↗
arXiv2023

Genuinely nonabelian partial difference sets

Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of \textit{genuinely nonabelian} PDSs, i.e., PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which -- one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil \cite{Godsil_1992} -- are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.

math.CO↗
arXiv2022

Null ideals of sets of $3 \times 3$ similar matrices with irreducible characteristic polynomial

Let $F$ be a field and $M_n(F)$ the ring of $n \times n$ matrices over $F$. Given a subset $S$ of $M_n(F)$, the null ideal of $S$ is the set of all polynomials $f$ with coefficients from $M_n(F)$ such that $f(A) = 0$ for all $A \in S$. We say that $S$ is core if the null ideal of $S$ is a two-sided ideal of the polynomial ring $M_n(F)[x]$. We study sufficient conditions under which $S$ is core in the case where $S$ consists of $3 \times 3$ matrices, all of which share the same irreducible characteristic polynomial. In particular, we show that if $F$ is finite with $q$ elements and $|S| \geqslant q^3-q^2+1$, then $S$ is core. As a byproduct of our work, we obtain some results on block Vandermonde matrices, invertible matrix commutators, and graphs defined via an invertible difference relation.

math.RA↗
arXiv2022

Cliques and independent subgroups of the Birkhoff polytope graph

The Birkhoff polytope $\Omega_n$ is the polytope of doubly stochastic matrices of order $n$. The Birkhoff polytope graph $G(\Omega_n)$ is the skeleton of $\Omega_n$; it is the Cayley graph whose vertex set consists of the elements of the symmetric group ${\rm Sym}(n)$ of degree $n$, where two permutations are adjacent if one equals the product of the other with a cycle. We study the combinatorial structure of this graph, focusing on its maximal and maximum cliques and on its independent subgroups (subgroups of ${\rm Sym}(n)$ whose elements are pairwise nonadjacent in the graph). We obtain maximal subgroups of $G(\Omega_n)$ and establish both a lower bound and an upper bound for its clique number. Especially, we prove that if $K$ is a subset of ${\rm Sym}(n)$ consisting of 3-cycle permutations such that $\delta_1^{-1}\delta_2$ is a single cycle for all $\delta_1,\delta_2\in K$, then the maximum size of $K$ is $\lfloor (n-1)^2/4\rfloor$, which can be viewed as an Erd\H{o}s-Ko-Rado-type theorem for ${\rm Sym}(n)$.

math.CO↗
arXiv2022

A new infinite family of $\sigma$-elementary rings

A cover of an associative (not necessarily commutative nor unital) ring $R$ is a collection of proper subrings of $R$ whose set-theoretic union equals $R$. If such a cover exists, then the covering number $\sigma(R)$ of $R$ is the cardinality of a minimal cover, and a ring $R$ is called $\sigma$-elementary if $\sigma(R) < \sigma(R/I)$ for every nonzero two-sided ideal $I$ of $R$. In this paper, we provide the first examples of $\sigma$-elementary rings $R$ that have nontrivial Jacobson radical $J$ with $R/J$ noncommutative, and we determine the covering numbers of these rings.

math.RA↗
arXiv2022

Locally s-arc-transitive graphs arising from product action

We study locally $s$-arc-transitive graphs arising from the quasiprimitive product action (PA). We prove that, for any locally $(G,2)$-arc-transitive graph with $G$ acting quasiprimitively with type PA on both $G$-orbits of vertices, the group $G$ does not act primitively on either orbit. Moreover, we construct the first examples of locally $s$-arc-transitive graphs of PA type that are not standard double covers of $s$-arc-transitive graphs of PA type, answering the existence question for these graphs.

math.CO↗
arXiv2021

The covering numbers of rings

A cover of an associative (not necessarily commutative nor unital) ring $R$ is a collection of proper subrings of $R$ whose set-theoretic union equals $R$. If such a cover exists, then the covering number $\sigma(R)$ of $R$ is the cardinality of a minimal cover, and a ring $R$ is called $\sigma$-elementary if $\sigma(R) < \sigma(R/I)$ for every nonzero two-sided ideal $I$ of $R$. If $R$ is a ring with unity, then we define the unital covering number $\sigma_u(R)$ to be the size of a minimal cover of $R$ by subrings that contain $1_R$ (if such a cover exists), and $R$ is $\sigma_u$-elementary if $\sigma_u(R) < \sigma_u(R/I)$ for every nonzero two-sided ideal of $R$. In this paper, we classify all $\sigma$-elementary unital rings and determine their covering numbers. Building on this classification, we are further able to classify all $\sigma_u$-elementary rings and prove $\sigma_u(R) = \sigma(R)$ for every $\sigma_u$-elementary ring $R$. We also prove that, if $R$ is a ring without unity with a finite cover, then there exists a unital ring $R'$ such that $\sigma(R) = \sigma_u(R')$, which in turn provides a complete list of all integers that are the covering number of a ring. Moreover, if \[\mathscr{E}(N) := \{m : m \le N, \sigma(R) = m \text{ for some ring } R\},\] then we show that $|\mathscr{E}(N)| = \Theta(N/\log(N))$, which proves that almost all integers are not covering numbers of a ring.

math.RA↗
arXiv2020

Covering numbers of commutative rings

A cover of a unital, associative (not necessarily commutative) ring $R$ is a collection of proper subrings of $R$ whose set-theoretic union equals $R$. If such a cover exists, then the covering number $\sigma(R)$ of $R$ is the cardinality of a minimal cover, and a ring $R$ is called $\sigma$-elementary if $\sigma(R) < \sigma(R/I)$ for every nonzero two-sided ideal $I$ of $R$. In this paper, we show that if $R$ has a finite covering number, then the calculation of $\sigma(R)$ can be reduced to the case where $R$ is a finite ring of characteristic $p$ and the Jacobson radical $J$ of $R$ has nilpotency 2. Our main result is that if $R$ has a finite covering number and $R/J$ is commutative (even if $R$ itself is not), then either $\sigma(R)=\sigma(R/J)$, or $\sigma(R)=p^d+1$ for some $d \geqslant 1$. As a byproduct, we classify all commutative $\sigma$-elementary rings with a finite covering number and characterize the integers that occur as the covering number of a commutative ring.

math.RA↗
arXiv2020

Restrictions on parameters of partial difference sets in nonabelian groups

A partial difference set $S$ in a finite group $G$ satisfying $1 \notin S$ and $S = S^{-1}$ corresponds to an undirected strongly regular Cayley graph ${\rm Cay}(G,S)$. While the case when $G$ is abelian has been thoroughly studied, there are comparatively few results when $G$ is nonabelian. In this paper, we provide restrictions on the parameters of a partial difference set that apply to both abelian and nonabelian groups and are especially effective in groups with a nontrivial center. In particular, these results apply to $p$-groups, and we are able to rule out the existence of partial difference sets in many instances.

math.CO↗
arXiv2019

Homomorphisms of matrix algebras and constructions of Butson-Hadamard matrices

An $n \times n$ matrix $H$ is Butson-Hadamard if its entries are $k^{\text{th}}$ roots of unity and it satisfies $HH^* = nI_n$. Write $BH(n, k)$ for the set of such matrices. Suppose that $k = p^{\alpha}q^{\beta}$ where $p$ and $q$ are primes and $\alpha \geq 1$. A recent result of {\"O}sterg{\aa}rd and Paavola uses a matrix $H \in BH(n,pk)$ to construct $H' \in BH(pn, k)$. We simplify the proof of this result and remove the restriction on the number of prime divisors of $k$. More precisely, we prove that if $k = mt$, and each prime divisor of $k$ divides $t$, then we can construct a matrix $H' \in BH(mn, t)$ from any $H \in BH(n,k)$.

math.CO↗
arXiv2019

Fuchs' problem for 2-groups

Nearly $60$ years ago, L\'{a}szl\'{o} Fuchs posed the problem of determining which groups can be realized as the group of units of a commutative ring. To date, the question remains open, although significant progress has been made. Along this line, one could also ask the more general question as to which finite groups can be realized as the group of units of a finite ring. In this paper, we consider the question of which $2$-groups are realizable as unit groups of finite rings, a necessary step toward determining which nilpotent groups are realizable. We prove that all $2$-groups of exponent $4$ and exponent $2$ are realizable in characteristic $2$, and we prove that many $2$-groups with exponent $4$ and nilpotency class $3$ are realizable in characteristic $2$. On the other hand, we provide an example of a $2$-group with exponent $4$ and nilpotency class $4$ that is not realizable in characteristic $2$. Moreover, while some groups of exponent greater than $4$ are realizable as unit groups of rings, we prove that any $2$-group with a self-centralizing element of order $8$ or greater is never realizable in characteristic $2^m$, and consequently any indecomposable, nonabelian group with a self-centralizing element of order $8$ or greater cannot be the group of units of a finite ring.

math.RA↗