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Ryan McCulloch

Publications and source records attributed to Ryan McCulloch.

18 recordsLinked to original sources
arXiv2026

Some answers regarding factorizations of finite groups

In this note we answer some questions of G. M. Bergman concerning factorizations of finite groups. Such questions originated when M. H. Hooshmand asked whether, given a finite group $G$ and a factorization $|G| = n_1 \cdots n_k$, one can always find subsets $A_1, \dots , A_k$ of $G$ with $|A_i| = n_i$ such that $G = A_1 \cdots A_k$. This was Question 19.35 of the Kourovka Notebook. G. M. Bergman provided a counterexample for $k=3$, and Kabenyuk provided counterexamples for all $k \geq 3$. We present these counterexamples with direct proofs that mirror Bergman's original $k=3$ argument, and we answer another recent question of Bergman.

math.GR↗
arXiv2026

The category of centralizer lattices of groups

We formalize the concept of a centralizer-respecting homomorphism, surjective homomorphisms which are equivariant with respect to taking the centralizer of a subgroup. There is a functor from the category of centralizer-respecting homomorphisms to the category of centralizer lattices. Finally, we conclude with some theorems about centralizer-respecting homomorphisms that show that the category of centralizer-respecting homomorphisms has many interesting maps.

math.GR↗
arXiv2026

An answer regarding automorphisms of finite abelian groups

In this note we provide a negative answer to the question: ``Is it true that for every positive rational number $r$ there exists a finite abelian group $G$ such that $|\mathrm{Aut}(G)|/|G| = r$?". We show that if $r = a/b$ is a rational number (with $a$ and $b$ coprime integers) so that $r = |\mathrm{Aut}(G)|/|G|$ for a finite abelian group $G$, then $b$ is squarefree. We also show that no odd prime can equal $ |\mathrm{Aut}(G)|/|G|$ for a finite abelian group $G$.

math.GR↗
arXiv2026

Finite groups with many elements of the same order

We study a conjecture by Deaconescu on the solubility of finite groups with claims that if more than half of the elements in a finite group has the same order $k$, then the group is soluble. We show that the original conjecture fails by presenting some counterexamples. By restricting to a fixed $k$, the conjecture may or may not hold depending on $k$. We prove that if $k$ is a power of a prime other than $2$ or $3$, or if $k=2, 3$ or $4$, then the conjecture holds, while it fails for many other choices of $k$ including all multiples of $2$ and $3$ which are larger than $5$. For $k=4$ we also find the sharp upper bound of the ratio of elements of order $4$ in non-soluble groups. We also prove that for all $k>1$, it is always possible to find a finite non-soluble group where at least $2/15$ of the elements have order $k$.

math.GR↗
arXiv2025

Covering by Centralizers

In this paper, we consider covers of finite groups by centralizers of elements. We show that the set of centralizers that are maximal under the partial ordering form a cover of the group. We also show that the set of centralizers that are minimal under the partial ordering form a cover of the group. We show for $F$-groups that are nonabelian $p$-groups that the number of distinct nontrivial centralizers is congruent to $1$ modulo $p$.

math.GR↗
arXiv2025

The commuting graph and a graph associated with centralizers

Let $G$ be a $p$-group. We begin to consider the relationship between the structure of the commuting graph and $|G:Z(G)|$. We also build a family of groups whose commuting graphs have more than one connected component whose diameter is at least $2$. For this, we introduce another graph related to the commuting graph that is associated with centralizers.

math.GR↗
arXiv2025

The Cycle Counts of Graphs

We prove that an inseparable graph can have any positive number of cycles with the six exceptions 2, 4, 5, 8, 9, 16, and that an inseparable cubic graph has the additional exceptions 1 and 13. The exceptions for simple inseparable cubic graphs are unknown.

math.CO↗
arXiv2025

On the Chermak-Delgado lattice of a finite group

By imposing conditions upon the index of a self-centralizing subgroup of a group, and upon the index of the center of the group, we are able to classify the Chermak-Delgado lattice of the group. This is our main result. We use this result to classify the Chermak-Delgado lattices of dicyclic groups and of metabelian $p$-groups of maximal class.

math.GR↗
arXiv2025

Finite groups with dense ${\cal CD}$-subgroups

A group $G$ is said to have dense ${\cal CD}$-subgroups if each non-empty open interval of the subgroup lattice $L(G)$ contains a subgroup in the Chermak--Delgado lattice ${\cal CD}(G)$. In this note, we study finite groups satisfying this property.

math.GR↗
arXiv2025

Incidence Gain Graphs and Generalized Quadrangles

We demonstrate a construction method based on a gain function that is defined on the incidence graph of an incidence geometry. Restricting to when the incidence geometry is a linear space, we show that the construction yields a generalized quadrangle provided that the gain function satisfies a certain bijective property. Our method is valid for finite and infinite geometries. We produce a family of generalized quadrangles by defining such a gain function on an affine plane over an arbitrary field.

math.CO↗
arXiv2024

Locally resolvable BIBDs and generalized quadrangles with ovoids

In this note we establish a 1-to-1 correspondence between the class of generalized quadrangles with ovoids and the class of balanced incomplete block designs that posses a non-triangular local resolution system and have the appropriate parameters. We present a non-triangular local resolution system for a difference family BIBD construction of Sprott.

math.CO↗
arXiv2024

The Chermak-Delgado Measure as a Map on Posets

The Chermak-Delgado measure of a finite group is a function which assigns to each subgroup a positive integer. In this paper, we give necessary and sufficient conditions for when the Chermak-Delgado measure of a group is actually a map of posets, i.e., a monotone function from the subgroup lattice to the positive integers. We also investigate when the Chermak-Delgado measure, restricted to the centralizers, is increasing.

math.GR↗
arXiv2022

On groups with few subgroups not in the Chermak-Delgado lattice

We investigate the question of how many subgroups of a finite group are not in its Chermak-Delgado lattice. The Chermak-Delgado lattice for a finite group is a self-dual lattice of subgroups with many intriguing properties. Fasol\u{a} and T\u{a}rn\u{a}uceanu asked how many subgroups are not in the Chermak-Delgado lattice and classified all groups with two or less subgroups not in the Chermak-Delgado lattice. We extend their work by classifying all groups with less than five subgroups not in the Chermak-Delgado lattice. In addition, we show that a group with less than five subgroups not in the Chermak--Delgado lattice is nilpotent. In this vein we also show that the only non-nilpotent group with five or fewer subgroups in the Chermak-Delgado lattice is S_3.

math.GR↗
arXiv2022

Central Products and the Chermak-Delgado Lattice

The Chermak-Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups. We prove that the Chermak-Delgado lattice of a central product contains the product of the Chermak-Delgado lattices of the relevant central factors. Furthermore, we obtain information about heights of elements in the Chermak-Delgado lattice relative to their heights in the Chermak-Delgado lattices of central factors. We also explore how the central product can be used as a tool in investigating Chermak-Delgado lattices.

math.GR↗
arXiv2018

Two classes of finite groups whose Chermak-Delgado lattice is a chain of length zero

It is an open question in the study of Chermak-Delgado lattices precisely which finite groups $G$ have the property that $CD(G)$ is a chain of length $0$. In this note, we determine two classes of groups with this property. We prove that if $G=AB$ is a finite group, where $A$ and $B$ are abelian subgroups of relatively prime orders with $A$ normal in $G$, then the Chermak-Delgado lattice of $G$ equals $\{AC_B(A)\}$, a strengthening of earlier known results.

math.GR↗
arXiv2017

Finite Groups with a Trivial Chermak-Delgado Subgroup

The Chermak-Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups of $G$. The least element of the Chermak-Delgado lattice of $G$ is known as the Chermak-Delgado subgroup of $G$. This paper concerns groups with a trivial Chermak-Delgado subgroup. We prove that if the Chermak-Delgado lattice of such a group is lattice isomorphic to a Cartesian product of lattices, then the group splits as a direct product, with the Chermak-Delgado lattice of each direct factor being lattice isomorphic to one of the lattices in the Cartesian product. We establish many properties of such groups and properties of subgroups in the Chermak-Delgado lattice. We define a CD-minimal group to be an indecomposable group with a trivial Chermak-Delgado subgroup. We establish lattice theoretic properties of Chermak-Delgado lattices of CD-minimal groups. We prove an extension theorem for CD-minimal groups, and use the theorem to produce twelve examples of CD-minimal groups, each having different CD lattices. Curiously, quasi-antichain $p$-group lattices play a major role in the author's constructions.

math.GR↗
arXiv2015

Chermak-Delgado Simple Groups

This paper provides the first steps in classifying the finite solvable groups having Property A, which is a property involving abelian normal subgroups. We see that this classification is reduced to classifying the solvable Chermak-Delgado simple groups, which the author defines. The author completes a classification of Chermak-Delgado simple groups under certain restrictions on the primes involved in the group order.

math.GR↗