Searcharxiv⌕ Search

arXiv subjects

Hung Ngoc Nguyen

Publications and source records attributed to Hung Ngoc Nguyen.

At least 19 recordsLinked to original sources
arXiv2015

Variations of Landau's theorem for p-regular and p-singular conjugacy classes

The well-known Landau's theorem states that, for any positive integer $k$, there are finitely many isomorphism classes of finite groups with exactly $k$ (conjugacy) classes. We study variations of this theorem for $p$-regular classes as well as $p$-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of $p$-regular classes or the number of $p$-singular classes of the group. In particular, if $G$ is a finite group with $O_p(G)=1$ then $|G/F(G)|_{p'}$ is bounded in terms of the number of $p$-regular classes of $G$. However, it is not possible to prove that there are finitely many groups with no nontrivial normal $p$-subgroup and $k$ $p$-regular classes without solving some extremely difficult number-theoretic problems (for instance, we would need to show that the number of Fermat primes is finite).

math.GR↗
arXiv2015

Complex group algebras of the double covers of the symmetric and alternating groups

We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let $n\geq 5$ be an integer, $G$ a finite group, and let $\AAA$ and $\SSS^\pm$ denote the double covers of $\Al_n$ and $\Sy_n$, respectively. We prove that $\CC G\cong \CC \AAA$ if and only if $G\cong \AAA$, and $\CC G\cong \CC \SSS^+\cong\CC\SSS^-$ if and only if $G\cong \SSS^+$ or $\SSS^-$. This in particular completes the proof of a conjecture proposed by the second and fourth authors that every finite quasi-simple group is determined uniquely up to isomorphism by the structure of its complex group algebra. The known results on prime power degrees and relatively small degrees of irreducible (linear and projective) representations of the symmetric and alternating groups together with the classification of finite simple groups play an essential role in the proofs.

math.RT↗
arXiv2015

Projective special linear groups $PSL_4(q)$ are determined by the set of their character degrees

Let $G$ be a finite group and let $cd(G)$ be the set of all irreducible complex character degrees of $G$. It was conjectured by Huppert in Illinois J. Math. 44 (2000) that, for every non-abelian finite simple group $H$, if $cd(G)=cd(H)$ then $G\cong H\times A$ for some abelian group $A$. In this paper, we confirm the conjecture for the family of projective special linear groups $\textrm{PSL}_4(q)$ with $q\geq 13$.

math.GR↗
arXiv2014

The largest character degrees of the symmetric and alternating groups

We show that the largest character degree of an alternating group $A_n$ with $n\geq 5$ can be bounded in terms of smaller degrees in the sense that \[ b(A_n)^2<\sum_{ψ\in\textrm{Irr}(A_n),\,ψ(1)< b(A_n)}ψ(1)^2, \] where $\textrm{Irr}(A_n)$ and $b(A_n)$ respectively denote the set of irreducible complex characters of $A_n$ and the largest degree of a character in $\textrm{Irr}(A_n)$. This confirms a prediction of I. M. Isaacs for the alternating groups and answers a question of M. Larsen, G. Malle, and P. H. Tiep.

math.GR↗
arXiv2014

On a conjecture of Gluck

Let $F(G)$ and $b(G)$ respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group $G$. A well-known conjecture of D. Gluck claims that if $G$ is solvable then $|G:F(G)|\leq b(G)^{2}$. We confirm this conjecture in the case where $|F(G)|$ is coprime to 6. We also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure.

math.GR↗
arXiv2014

On the number of conjugacy classes of $π$-elements in finite groups

Let $G$ be a finite group and $π$ be a set of primes. We show that if the number of conjugacy classes of $π$-elements in $G$ is larger than $5/8$ times the $π$-part of $|G|$ then $G$ possesses an abelian Hall $π$-subgroup which meets every conjugacy class of $π$-elements in $G$. This extends and generalizes a result of W. H. Gustafson.

math.GR↗
arXiv2013

On the average character degree of finite groups

We prove that if the average of the degrees of the irreducible characters of a finite group $G$ is less than 16/5, then $G$ is solvable. This solves a conjecture of I.M. Isaacs, M. Loukaki, and the first author. We discuss related questions.

math.GR↗
arXiv2013

On the commuting probability and supersolvability of finite groups

For a finite group $G$, let $d(G)$ denote the probability that a randomly chosen pair of elements of $G$ commute. We prove that if $d(G)>1/s$ for some integer $s>1$ and $G$ splits over an abelian normal nontrivial subgroup $N$, then $G$ has a nontrivial conjugacy class inside $N$ of size at most $s-1$. We also extend two results of Barry, MacHale, and N\'ı Shé on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that if $d(G)>5/16$ then either $G$ is supersolvable, or $G$ isoclinic to $A_4$, or $G/\Center(G)$ is isoclinic to $A_4$.

math.GR↗
arXiv2013

Character degree sums of finite groups

We present some results on character degree sums in connection with certain characteristics of finite groups such as p-solvability, solvability, supersolvability, and nilpotency. Some of them strengthen known results in the literature.

math.GR↗
arXiv2013

Controlling composition factors of a finite group by its character degree ratio

For a finite nonabelian group $G$ let $\rat(G)$ be the largest ratio of degrees of two nonlinear irreducible characters of $G$. We show that nonabelian composition factors of $G$ are controlled by $\rat(G)$ in some sense. Specifically, if $S$ different from the simple linear groups $\PSL_2(q)$ is a nonabelian composition factor of $G$, then the order of $S$ and the number of composition factors of $G$ isomorphic to $S$ are both bounded in terms of $\rat(G)$. Furthermore, when the groups $\PSL_2(q)$ are not composition factors of $G$, we prove that $|G:\Oinfty(G)|\leq \rat(G)^{21}$ where $\Oinfty(G)$ denotes the solvable radical of $G$.

math.GR↗
arXiv2011

Quasisimple classical groups and their complex group algebras

Let $H$ be a finite quasisimple classical group, i.e. $H$ is perfect and $S:=H/Z(H)$ is a finite simple classical group. We prove in this paper that, excluding the cases when the simple group $S$ has a very exceptional Schur multiplier such as $\PSL_3(4)$ or $\PSU_4(3)$, $H$ is uniquely determined by the structure of its complex group algebra. The proofs make essential use of the classification of finite simple groups as well as the results on prime power character degrees and relatively small character degrees of quasisimple classical groups.

math.GR↗
arXiv2011

Multiplicities of conjugacy class sizes of finite groups

It has been proved recently by Moreto and Craven that the order of a finite group is bounded in terms of the largest multiplicity of its irreducible character degrees. A conjugacy class version of this result was proved for solvable groups by Zaikin-Zapirain. In this note, we prove that if $G$ is a finite simple group then the order of $G$, denoted by $|G|$, is bounded in terms of the largest multiplicity of its conjugacy class sizes and that if the largest multiplicity of conjugacy class sizes of any quotient of a finite group $G$ is $m$, then $|G|$ is bounded in terms of $m$.

math.GR↗
arXiv2010

On the permutation modules for orthogonal groups $O_{m}^{\pm}(3)$ acting on nonsingular points of their standard modules

We describe the structure, including composition factors and submodule lattices, of cross-characteristic permutation modules for the natural actions of the orthogonal groups $O_{m}^{\pm}(3)$ with $m\geq6$ on nonsingular points of their standard modules. These actions together with those studied in \cite{HN} are all examples of primitive rank 3 actions of finite classical groups on nonsingular points.

math.GR↗
arXiv2009

Low-dimensional complex characters of the symplectic and orthogonal groups

We classify the irreducible complex characters of the symplectic groups $Sp_{2n}(q)$ and the orthogonal groups $Spin_{2n}^\pm(q)$, $Spin_{2n+1}(q)$ of degrees up to the bound D, where $D=(q^n-1)q^{4n-10}/2$ for symplectic groups, $D=q^{4n-8}$ for orthogonal groups in odd dimension, and $D=q^{4n-10}$ for orthogonal groups in even dimension.

math.RT↗
arXiv2009

Irreducible restrictions of Brauer characters of the Chevalley group G_2(q) to its proper subgroups

Let $G_2(q)$ be the Chevalley group of type $G_2$ defined over a finite field with q=p^n elements, where p is a prime number and $n$ is a positive integer. In this paper, we determine when the restriction of an absolutely irreducible representation of $G$ in characteristic other than p to a maximal subgroup of $G_2(q)$ is still irreducible. Similar results are obtained for $^2B_2(q)$ and $^2G_2(q)$.

math.RT↗