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Mark L. Lewis

Publications and source records attributed to Mark L. Lewis.

At least 73 records · Page 4Linked to original sources
arXiv2020

Groups that have a Partition by Commuting Subsets

Let $G$ be a nonabelian group. We say that $G$ has an abelian partition, if there exists a partition of $G$ into commuting subsets $A_1, A_2, \ldots, A_n$ of $G$, such that $|A_i|\geqslant 2$ for each $i=1, 2, \ldots, n$. This paper investigates problems relating to group with abelian partitions. Among other results, we show that every finite group is isomorphic to a subgroup of a group with an abelian partition and also isomorphic to a subgroup of a group with no abelian partition. We also find bounds for the minimum number of partitions for several families of groups which admit abelian partitions -- with exact calculations in some cases. Finally, we examine how the size of partitions with the minimum number of parts behaves with respect to the direct product.

math.GR↗
arXiv2020

On solvable groups with one vanishing class size

Let $G$ be a finite group, and let cs$(G)$ be the set of conjugacy class sizes of $G$. Recalling that an element $g$ of $G$ is called a \emph{vanishing element} if there exists an irreducible character of $G$ taking the value $0$ on $g$, we consider one particular subset of cs$(G)$, namely, the set vcs$(G)$ whose elements are the conjugacy class sizes of the vanishing elements of $G$. Motivated by the results in \cite{BLP}, we describe the class of the finite groups $G$ such that vcs$(G)$ consists of a single element \emph{under the assumption that $G$ is supersolvable or $G$ has a normal Sylow $2$-subgroup} (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size \emph{which is either a prime power or square-free}.

math.GR↗
arXiv2020

Toward a Classification of the Supercharacter Theories of $C_p\times C_p$

In this paper, we study the superscharacter theories of elementary abelian $p$-groups of order $p^2$. We show that the supercharacter theories that arise from the direct product construction and the $\ast$-product construction can be obtained from automorphisms. We also prove that any supercharacter theory of an elementary abelian $p$-group of order $p^2$ that has a nonidentity superclass of size $1$ or a nonprincipal linear supercharacter must come from either a $\ast$-product or a direct product. Although we are unable to prove results for general primes, we do compute all of the supercharacter theories when $p = 2, 3, 5$, and based on these computations along with particular computations for larger primes, we make several conjectures for a general prime $p$.

math.GR↗
arXiv2020

The cyclic graph (deleted enhanced power graph) of a direct product

Let $G$ be a finite group. Define a graph on the set $G^{\#} = G \setminus \{ 1 \}$ by declaring distinct elements $x,y\in G^{\#}$ to be adjacent if and only if $\langle x,y\rangle$ is cyclic. Denote this graph by $Δ(G)$. The graph $Δ(G)$ has appeared in the literature under the names cyclic graph and deleted enhanced power graph. If $G$ and $H$ are nontrivial groups, then $Δ(G\times H)$ is completely characterized. In particular, if $Δ(G\times H)$ is connected, then a diameter bound is obtained, along with an example meeting this bound. Also, necessary and sufficient conditions for the disconnectedness of $Δ(G\times H)$ are established.

math.GR↗
arXiv2020

The Solubility Graph Associated With a Finite Group

Let $G$ be a finite group. The solubility graph associated with the finite group $G$, denoted by $Γ_{\cal S}(G)$, is a simple graph whose vertices are the non-trivial elements of $G$, and there is an edge between two distinct elements $x$ and $y$ if and only if $\langle x, y\rangle$ is a soluble subgroup of $G$. In this paper, we examine some properties of solubility graphs.

math.GR↗
arXiv2019

Characterizations of nested GVZ-groups by central series

Many properties of groups can be defined by the existence of a particular normal series. The classic examples being solvability, supersolvability and nilpotence. Among the nilpotent groups are the so-called nested GVZ-groups --- groups where the centers of the irreducible characters form a chain, and where every irreducible character vanishes off of its center. In this paper, we show that nested GVZ-groups can be characterized by the existence of a certain ascending central series, or by the existence of a certain descending central series.

math.GR↗
arXiv2019

Vanishing-off subgroups and supercharacter theory products

In this paper, we study the vanishing-off subgroups of supercharacters, and use these to determine several new characterizations of supercharacter theory products. In particular, we give a character theoretic characterization that allows us to conclude that one may determine if a supercharacter theory is a $Δ$-product or $\ast$-product from the values of its corresponding supercharacters.

math.RT↗
arXiv2019

A Generalization of the Hughes Subgroup

Let $G$ be a finite group, $π$ be a set of primes, and define $H_π(G)$ to be the subgroup generated by all elements of $G$ which do not have prime order for every prime in $π$. In this paper, we investigate some basic properties of $H_π(G)$ and its relationship to the Hughes subgroup. We show that for most groups, only one of three possibilities occur: $H_π(G) = 1$, $H_π(G)=G$, or $H_π(G) = H_{p}(G)$ for some prime $p \in π$. There is only one other possibility: $G$ is a Frobenius group whose Frobenius complement has prime order $p$, and whose Frobenius kernel, $F$, is a nonabelian $q$-group such that $H_π(G)$ arises as a proper and nontrivial Hughes subgroup of $F$. We investigate a few restrictions on the possible choices of the primes $p$ and $q$.

math.GR↗
arXiv2019

Groups where the centers of the irreducible characters form a chain

We consider groups where the centers of the irreducible characters form a chain. We obtain two alternate characterizations of these groups, and we obtain some information regarding the structure of these groups. Using our results, we are able to classify those groups where the kernels of the irreducible characters form a chain. We show that a result of Nenciu regarding nested GVZ groups is really a result about nested groups. We obtain an alternate proof of a theorem of Isaacs regarding the existence of $p$-groups with a given set of irreducible character degrees.

math.GR↗
arXiv2019

$p$-groups with exactly four codegrees

Let $G$ be a $p$-group and let $χ$ be an irreducible character of $G$. The codegree of $χ$ is given by $|G:\text{ker}(χ)|/χ(1)$. Du and Lewis have shown that a $p$-group with exactly three codegrees has nilpotence class at most 2. Here we investigate $p$-groups with exactly four codegrees. If, in addition to having exactly four codegrees, $G$ has two irreducible character degrees, $G$ has largest irreducible character degree $p^2$, $|G:G'|=p^2$, or $G$ has coclass at most 3, then $G$ has nilpotence class at most 4. In the case of coclass at most 3, the order of $G$ is bounded by $p^7$. With an additional hypothesis we can extend this result to $p$-groups with four codegrees and coclass at most 7. In this case the order of $G$ is bounded by $p^{11}$.

math.GR↗
arXiv2018

$p$-groups with $p^2$ as a codegree

Let $G$ be a $p$-group and let $χ$ be an irreducible character of $G$. The codegree of $χ$ is given by $|G:\text{ker}(χ)|/χ(1)$. This paper investigates the relationship between the nilpotence class of a group and the inclusion of $p^2$ as a codegree. If $G$ is a finite $p$-group with coclass $2$ and order at least $p^5$, or coclass $3$ and order at least $p^6$, then $G$ has $p^2$ as a codegree. With an additional hypothesis this result can be extended to $p$-groups with coclass $n\ge 3$ and order at least $p^{2n}$.

math.GR↗
arXiv2018

Simple Groups Whose Prime Graph or Solvable Graph is Split

A graph is split if there is a partition of its vertex set into a clique and an independent set. The present paper is devoted to the splitness of some graphs related to finite simple groups, namely, prime graphs and solvable graphs, and their compact forms. It is proved that the compact form of the prime graph of any finite simple group is split.

math.GR↗
arXiv2018

Character codegrees of maximal class p-groups

Let $G$ be a $p$-group and let $χ$ be an irreducible character of $G$. The codegree of $χ$ is given by $|G:\text{ker}(χ)|/χ(1)$. If $G$ is a maximal class $p$-group that is normally monomial or has at most three character degrees then the codegrees of $G$ are consecutive powers of $p$. If $|G|=p^n$ and $G$ has consecutive $p$-power codegrees up to $p^{n-1}$ then the nilpotence class of $G$ is at most 2 or $G$ has maximal class.

math.GR↗
arXiv2018

Enumerating isoclinism classes of semi-extraspecial groups

We enumerate the number of isoclinism classes of semi-extraspecial $p$-groups with derived subgroup of order $p^2$. To do this, we enumerate $\text{GL}(2, p)$-orbits of sets of irreducible, monic polynomials in $\mathbb{F}_p[x]$. Along the way, we also provide a new construction of an infinite family of semi-extraspecial groups as central quotients of Heisenberg groups over local algebras.

math.GR↗
arXiv2018

Classifying Character Degree Graphs With 6 Vertices

We investigate prime character degree graphs of solvable groups that have six vertices. There are one hundred twelve non-isomorphic connected graphs with six vertices, of which all except nine are classified in this paper. We also completely classify the disconnected graphs with six vertices.

math.GR↗