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Mark L. Lewis

Publications and source records attributed to Mark L. Lewis.

At least 19 recordsLinked to original sources
arXiv2026

Constructing solvable groups whose character degree graphs generalize the bowtie

We present here a generalized construction of a finite solvable group whose prime character degree graph has the shape and structure of the bowtie graph. As with the original bowtie, the graphs obtained by this generalized construction, under certain restrictions, cannot be realized by the usual method of taking direct products of smaller graphs. Within the condition of $n=1$, we show how this recovers the original bowtie graph, which has five vertices. We also provide examples and explicit choices of primes which generate graphs with more vertices.

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

Groups with a conjugacy class that is the difference of two normal subgroups

We consider finite groups having a conjugacy class that is the difference of two normal subgroups. That is, suppose $G$ is a group and $M$ and $N$ are normal subgroups so that $N < M$, and suppose that there is an element $g \in G$ so that the conjugacy class of $g$ is $M \setminus N$. We find a character-theoretic characterization of this condition, and we determine some structural properties of groups with such a conjugacy class. If we add the condition that $M/N$ is the unique minimal normal subgroup of $G/N$, then we obtain a generalization of a result by S.M. Gagola.

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arXiv2026

Classifying Prime Character Degree Graphs With Eight Vertices

In this paper, an effort is made to classify which prime character degree graphs having eight vertices occur for some finite solvable group. To approach this, we compile known results and constructions from the literature which are used to develop a general algorithm to begin classifying graphs of any order. We then apply the algorithm to the graphs of order eight. Of the 12,346 non-isomorphic graphs with eight vertices, 1,229 are disconnected and are fully classified. Meanwhile, 37 of the 11,117 non-isomorphic connected graphs are shown to occur; 34 of which are constructed via direct products and 3 of which have diameter three. Fifty-six graphs are shown not to occur, several of which fall into previously studied families, while the classification of 206 graphs is still unknown.

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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

Sylow subgroups and the number of irreducible characters of degrees divisible by a prime $p$

Let $G$ be a finite group and $p$ a prime. We establish an upper bound for the derived length of a Sylow $p$-subgroup of $G$ in terms of the number of irreducible characters of $G$ whose degrees are divisible by $p$. We also prove that if $B$ is a $p$-block of a finite $p$-solvable group $G$ with defect group $D$, then the derived length of $D$ is at most one more than the number of ordinary irreducible characters of positive height in $B$.

math.GR↗
arXiv2025

Groups Having a Character of Maximal Degree

Let $G$ be a group, let $d$ be a character degree, and let $e$ be the integer so that $|G| = d(d+e)$. It has been shown when $e > 1$ that $|G| \le e^4 - e^3$. In this paper, we consider the groups where $|G| = e^4 - e^3$. It is known that $e$ must be a power of a prime. We classify the groups where $e$ is a prime and where $e$ is $4$, $9$, and $25$. In so doing, we find a new nonsolvable Camina pair.

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

A converse for a theorem of Gallagher

Let $G$ be a finite group. Suppose $N$ is a normal subgroup of $G$. Recall that Gallagher's theorem states that if $\chi \in {\rm Irr} (G)$ satisfies $\chi_N$ is irreducible, then $\chi \beta$ is irreducible and distinct for all $\beta \in {\rm Irr} (G/N)$. Furthermore, if $\theta = \chi_N$, then these are all of the irreducible constituents of $\theta^G$. We prove that the converse of this theorem holds. We also prove that a partial converse of the Brauer version of this theorem holds. Finally, we prove that an analog of Gallagher's theorem holds for Isaacs' $\pi$-partial characters and that a partial converse of that theorem is true.

math.GR↗
arXiv2025

Isaacs' Generalization of Taketa's theorem

We generalize the definition of pseudo monomial characters and $M$-groups to the Brauer character and Isaacs' $\pi$-partial character settings. We prove an analogs of Isaacs's generalization of Taketa's theorem in those settings. We consider other analogs of results regarding $M$-groups in those settings.

math.GR↗
arXiv2025

Group cosets with all elements of equal order

Let $G$ be a finite group and $N$ a proper, nontrivial, normal subgroup of $G$. If, for every element $x$ of $G$ not lying in $N$, the elements in the coset $xN$ all have the same order as $x$, then we say that $(G,N)$ is an {\it{equal order pair}}. This generalizes the concept of a Camina pair, that was introduced by the first author. In the present paper we study several properties of equal order pairs, showing that in many respects they resemble Camina pairs, but with some important differences.

math.GR↗
arXiv2025

On the Number of Disconnected Character Degree Graphs Satisfying P\'alfy's Inequality

Let $G$ be a finite solvable group with disconnected character degree graph $\Delta(G)$. Under these conditions, it follows from a result of P\'alfy that $\Delta(G)$ consists of two connected components. Another result of P\'alfy's gives an inequality relating the sizes of these two connected components. In this paper, we calculate the number of possible component size pairs that satisfy P\'alfy's inequality. Additionally, for a fixed positive integer $n$, the number of distinct graph orders for which exactly $n$ component size pairs satisfy P\'alfy's inequality is shown.

math.CO↗
arXiv2025

$M$-groups and Codegrees; $M_{p}$-groups and Brauer Character Degrees

Let $G$ be a finite group and $p$ be a prime. We prove that if $G$ has three codegrees, then $G$ is an $M$-group. We prove for some prime $p$ that if every irreducible Brauer character of $G$ is a prime, then for every normal subgroup $N$ of $G$ either $G/N$ or $N$ is an $M_p$-group.

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