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Iulian I. Simion

Publications and source records attributed to Iulian I. Simion.

8 recordsLinked to original sources
arXiv2025

More on Landau's theorem and Conjugacy Classes

In this paper we present two new results on the number of certain conjugacy classes of a finite group. For a finite group $G$, let $n(G)$ be the maximum of $k_{p}(G)$ taken over all primes $p$ where $k_{p}(G)$ denotes the number of conjugacy classes of nontrivial $p$-elements in $G$. Using a recent theorem of Giudici, Morgan and Praeger, we prove that there exists a function $f(x)$ with $f(x) \to \infty$ as $x \to \infty$ such that $n(G) \geq f(|G|)$ for any finite group $G$. Let $G$ be a finite group, and let $p$ be a prime dividing $|G|$. Let $k_{p'}(G)$ denote the number of conjugacy classes of elements of $G$ whose orders are coprime to $p$. We show that either $p=11$ and $G=C_{11}^2\rtimes \text{\rm SL}(2,5)$, or there exists a factorization $p-1 = ab$ with $a$ and $b$ positive integers, such that $k_{p}(G) \geq a$ and $k_{p'}(G) \geq b$ with equalities in both cases if and only if $G=C_p \rtimes C_b$ with $C_G(C_p) = C_p$.

math.GR↗
arXiv2020

Bounding the number of classes of a finite group in terms of a prime

Héthelyi and Külshammer showed that the number of conjugacy classes $k(G)$ of any solvable finite group $G$ whose order is divisible by the square of a prime $p$ is at least $(49p+1)/60$. Here an asymptotic generalization of this result is established. It is proved that there exists a constant $c>0$ such that for any finite group $G$ whose order is divisible by the square of a prime $p$ we have $k(G) \geq cp$.

math.GR↗
arXiv2018

On refined Bruhat decompositions and endomorphism algebras of Gelfand-Graev representations

Let $G$ be a finite reductive group defined over $\mathbb{F}_q$, with $q$ a power of a prime $p$. Motivated by a problem recently posed by C. Curtis, we first develop an algorithm to express each element of $G$ into a canonical form in terms of a refinement of a Bruhat decomposition, and we then use the output of the algorithm to explicitly determine the structure constants of the endomorphism algebra of a Gelfand-Graev representation of $G$ when $G=\mathrm{PGL}_3(q)$ for an arbitrary prime $p$, and when $G=\mathrm{SO}_5(q)$ for $p$ odd.

math.GR↗
arXiv2016

On small modules for quantum groups at roots of unity

A conjecture of De Concini Kac and Procesi provides a bound on the minimal possible dimension of an irreducible module for quantized enveloping algebras at an odd root of unity. We pose the problem of the existence of modules whose dimension equals this bound. We show that this question cannot have a positive answer in full generality and discuss variants of this question.

math.RT↗
arXiv2015

Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_{1}}\cdots A^{x_{k}}$ for $A\leq G$ and $x_{1},\ldots ,x_{k}\in G$, where $A$ is nilpotent or solvable. First we exploit the split $BN$-pair structure of finite simple groups of Lie type to give a unified self-contained proof that every such group is a product of four or three unipotent Sylow subgroups. Then we derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group. Finally, using conjugate factorizations of a general finite solvable group by any of its Carter subgroups, we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.

math.GR↗
arXiv2015

Groups equal to a product of three conjugate subgroups

Let $G$ be a finite non-solvable group. We prove that there exists a proper subgroup $A$ of $G$ such that $G$ is the product of three conjugates of $A$, thus replacing an earlier upper bound of $36$ with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group $G$ with a $BN$-pair and a finite Weyl group $W$ satisfies $G=\left( Bn_{0}B\right) ^{2}=BB^{n_{0}}B$ where $n_{0}$ is any preimage of the longest element of $W$. The proof of the last theorem is formulated in the dioid consisting of all unions of double cosets of $B$ in $G$. Other results on minimal length product covers of a group by conjugates of a proper subgroup are given.

math.GR↗