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Mark L. Lewis

Publications and source records attributed to Mark L. Lewis.

At least 91 records · Page 5Linked to original sources
arXiv2017

Semi-extraspecial groups with an abelian subgroup of maximal possible order

Let $p$ be a prime. A $p$-group $G$ is defined to be semi-extraspecial if for every maximal subgroup $N$ in $Z(G)$ the quotient $G/N$ is a an extraspecial group. In addition, we say that $G$ is ultraspecial if $G$ is semi-extraspecial and $|G:G'| = |G'|^2$. In this paper, we prove that every $p$-group of nilpotence class $2$ is isomorphic to a subgroup of some ultraspecial group. Given a prime $p$ and a positive integer $n$, we provide a framework to construct of all the ultraspecial groups order $p^{3n}$ that contain an abelian subgroup of order $p^{2n}$. In the literature, it has been proved that every ultraspecial group $G$ order $p^{3n}$ with at least two abelian subgroups of order $p^{2n}$ can be associated to a semifield. We provide a generalization of semifield, and then we show that every semi-extraspecial group $G$ that is the product of two abelian subgroups can be associated with this generalization of semifield.

math.GR↗
arXiv2017

Semi-extraspecial Groups

We survey the results regarding semi-extraspecial $p$-groups. Semi-extraspecial groups can be viewed as generalizations of extraspecial groups. We present the connections between semi-extraspecial groups and Camina groups and VZ-groups, and give upper bounds on the order of the center and the orders of abelian normal subgroups. We define ultraspecial groups to be semi-extraspecial groups where the center is as large as possible, and demonstrate a connection between ultraspecial groups that have at least two abelian subgroups whose order is the maximum and semifields.

math.GR↗
arXiv2017

Itô's theorem and monomial Brauer characters

Let $G$ be a finite solvable group, and let $p$ be a prime. In this note, we prove that $p$ does not divide $φ(1)$ for every irreducible monomial $p$-Brauer character $φ$ of $G$ if and only if $G$ has a normal Sylow $p$-subgroup.

math.GR↗
arXiv2017

Groups with one or two super-Brauer character theories

A super-Brauer character theory of a group $G$ and a prime $p$ is a pair consisting of a partition of the irreducible $p$-Brauer characters and a partition of the $p$-regular elements of $G$ that satisfy certain properties. We classify the groups and primes that have exactly one super-Brauer character theory. We discuss the groups with exactly two super-Brauer character theories.

math.GR↗
arXiv2017

Upper and lower bounds on Chillag table sums

Chillag has showed that there is a single generalization showing that the sums of ordinary character tables, Brauer character, and projective indecomposable characters are positive integers. We show that Chillag's construction also applies to Isaacs' $π$-partial characters. We show that if an extra condition is assumed, then we can obtain upper and lower bounds on the Chillag's table sums. We will demonstrate that this condition holds for Brauer characters, $π$-partial characters, and projective indecomposable characters, and so we obtain upper and lower bounds for the table sums in those cases. We also obtain results regarding the sums of rows and columns in these tables.

math.GR↗
arXiv2017

Groups satisfying the two-prime hypothesis with a composition factor isomorphic to ${\rm PSL}_2(q)$ for $q\geq 7$

Let $G$ be a finite group, and write ${\rm cd}(G)$ for the degree set of the complex irreducible characters of $G$. The group $G$ is said to satisfy the {\it two-prime hypothesis} if, for any distinct degrees $a, b \in {\rm cd}(G)$, the total number of (not necessarily different) primes of the greatest common divisor ${\rm gcd}(a, b)$ is at most $2$. In this paper, we prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to ${\rm PSL}_2 (q)$ for $q \geq 7$.

math.GR↗
arXiv2016

${\rm B}_π$-characters and quotients

Let $π$ be a set of primes, and let $G$ be a finite $π$-separable group. We consider the Isaacs ${\rm B}_π$-characters. We show that if $N$ is a normal subgroup of $G$, then ${\rm B}_π(G/N) = {\rm Irr} (G/N) \cap {\rm B}_π(G)$.

math.GR↗
arXiv2016

Brauer characters of $q^\prime$- degree

We show that if $p$ is a prime and $G$ is a finite $p$-solvable group satisfying the condition that a prime $q$ divides the degree of no irreducible $p$-Brauer character of $G$, then the normalizer of some Sylow $q$-subgroup of $G$ meets all the conjugacy classes of $p$-regular elements of $G$.

math.GR↗
arXiv2015

Centralizers of Camina $p$-groups of nilpotence class $3$

Let $G$ be a Camina $p$-group of nilpotence class $3$. We prove that if $G' < C_G (G')$, then $|Z(G)| \le |G':G_3|^{1/2}$. We also prove that if $G/G_3$ has only one or two abelian subgroups of order $|G:G'|$, then $G' < C_G (G')$. If $G/G_3$ has $p^a + 1$ abelian subgroups of order $|G:G'|$, then either $G' < C_G (G')$ or $|Z(G)| \le p^{2a}$.

math.GR↗
arXiv2015

Groups with exactly two supercharacter theories

In this paper, we classify those finite groups with exactly two supercharacter theories. We show that the solvable groups with two supercharacter theories are $\mathbb{Z}_3$ and $S_3$. We also show that the only nonsolvable group with two supercharacter theories is ${\rm Sp} (6,2)$.

math.GR↗
arXiv2015

Variations on average character degrees and $p$-nilpotence

We prove that if $p$ is an odd prime, $G$ is a solvable group, and the average value of the irreducible characters of $G$ whose degrees are not divisible by $p$ is strictly less than $2(p+1)/(p+3)$, then $G$ is $p$-nilpotent. We show that there are examples that are not $p$-nilpotent where this bound is met for every prime $p$. We then prove a number of variations of this result.

math.GR↗
arXiv2015

Landau's theorem, fields of values for characters, and solvable groups

When $G$ is solvable group, we prove that the number of conjugacy classes of elements of prime power order is less than or equal to the number of irreducible characters with values in fields where $\mathbb {Q}$ is extended by prime power roots of unity. We then combine this result with a theorem of Héthelyi and Külshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order to bound the order of a solvable group by the number of irreducible characters with values in fields extended by prime power roots of unity. This yields for solvable groups a generalization of Landau's theorem.

math.GR↗
arXiv2015

Finite groups with an irreducible character of large degree

Let $G$ be a finite group and $d$ the degree of a complex irreducible character of $G$, then write $|G|=d(d+e)$ where $e$ is a nonnegative integer. We prove that $|G|\leq e^4-e^3$ whenever $e>1$. This bound is best possible and improves on several earlier related results.

math.GR↗
arXiv2014

Camina $p$-groups that are generalized Frobenius complements

Let $P$ be a Camina $p$-group that acts on a group $Q$ in such a way that $C_P (x) \subseteq P'$ for all nonidentity elements $x \in Q$. We show that $P$ must be isomorphic to the quaternion group $Q_8$. If $P$ has class $2$, this is a known result, and this paper corrects a previously published erroneous proof of the general case.

math.GR↗
arXiv2014

Using a Galois connection to compute character degrees

Given a Mersenne prime $q$ and a positive even integer $e$, let $F$ and $E$ be the fields of orders $q$ and $q^e$ respectively. Let $C$ be a cyclic subgroup of $E^\times$ whose index in $E^\times$ is divisible only by primes dividing $q - 1$. We compute the character degrees of the group $C \rtimes {\rm Gal} (E/F)$ by using the Galois connection between the subfields of $E$ and the Galois group ${\rm Gal} (E/F)$.

math.GR↗