arXiv subjects

Claude Marion

Publications and source records attributed to Claude Marion.

15 recordsLinked to original sources

arXiv2025

On counting numerical semigroups by maximum primitive and Wilf's conjecture

We introduce a new way of counting numerical semigroups, namely by their maximum primitive, and show its relation with the counting of numerical semigroups by their Frobenius number. We show that these two ways of counting are M\"obius transforms of one another. We also establish that almost all numerical semigroups with large enough maximum primitive satisfy Wilf's conjecture. A crucial step in the proof is a result of independent interest: a numerical semigroup $S$ with multiplicity $\mathrm{m}$ such that $|S\cap (\mathrm{m},2 \mathrm{m})|\geq \sqrt{2\mathrm{m}}$ satisfies Wilf's conjecture.

math.CO↗

arXiv2024

On profinite groups with the Magnus Property

A group is said to have the Magnus Property (MP) if whenever two elements have the same normal closure then they are conjugate or inverse-conjugate. We show that a profinite MP group $G$ is prosolvable and any quotient of it is again MP. As corollaries we obtain that the only prime divisors of $|G|$ are $2$, $3$, $5$ and $7$, and the second derived subgroup of $G$ is pronilpotent. We also show that the inverse limit of an inverse system of profinite MP groups is again MP. Finally when $G$ is finitely generated, we establish that $G$ must in fact be finite.

math.GR↗

arXiv2023

On finite groups with the Magnus Property

We investigate finite groups with the Magnus Property, where a group is said to have the Magnus Property (MP) if whenever two elements have the same normal closure then they are conjugate or inverse conjugate. In particular we observe that a finite MP group is solvable, determine the finite primitive MP groups and determine all the possible orders of the chief factors of a finite MP group. We also determine the MP finite direct products of finite primitive groups, as well as the MP crown-based powers of a finite monolithic primitive group.

math.GR↗

arXiv2023

On the pseudovariety of groups $\mathbf{U} = \displaystyle\bigvee_{p \in \mathbb{P}} {\bf Ab}(p) \ast {\bf Ab}(p-1)$

We introduce the pseudovariety of finite groups $\mathbf{U} = \displaystyle\bigvee_{p \in \mathbb{P}} {\bf Ab}(p) \ast {\bf Ab}(p-1)$, where $\mathbb{P}$ is the set of all primes. We show that $\mathbf{U}$ consists of all finite supersolvable groups with elementary abelian derived subgroup and abelian Sylow subgroups, being therefore decidable. We prove that it is decidable whether or not a finitely generated subgroup of a free group is closed or dense for the pro-${\bf U}$ topology. We consider also the pseudovariety of finite groups ${\bf Ab}(p) \ast {\bf Ab}(d)$ (where $p$ is a prime and $d$ divides $p-1$). We study the pro-$({\bf Ab}(p) \ast {\bf Ab}(d))$ topology on a free group and construct the unique generator of minimum size of the pseudovariety ${\bf Ab}(p) \ast {\bf Ab}(d)$. Finally, we prove that the variety of groups generated by ${\bf U}$ is the variety of all metabelian groups, obtaining also results on the varieties generated by a Baumslag-Solitar group of the form $BS(1,q)$ for $q$ prime.

math.GR↗

arXiv2023

On the closure of cyclic subgroups of a free group in pro-V topologies

We determine the closure of a cyclic subgroup $H$ of a free group for the pro-{\bf V} topology when {\bf V} is an extension-closed pseudovariety of finite groups. We show that $H$ is always closed for the pro-nilpotent topology and compute its closure for the pro-$\mathbf{G}_p$ and pro-$\mathbf{V}_p$ topologies, where $\mathbf{G}_p$ and $\mathbf{V}_p$ denote respectively the pseudovariety of finite $p$-groups and the pseudovariety of finite groups having a normal Sylow $p$-subgroup with quotient an abelian group of exponent dividing $p-1$. More generally, given any nonempty set $P$ of primes, we consider the pseudovariety $\mathbf{G}_P$ of all finite groups having order a product of primes in $P$.

math.GR↗

arXiv2023

The pro-supersolvable topology on a free group: deciding denseness

Let $F$ be a free group of arbitrary rank and let $H$ be a finitely generated subgroup of $F$. Given a pseudovariety $\mathbf{V}$ of finite groups, i.e. a class of finite groups closed under taking subgroups, quotients and finitary direct products, we endow $F$ with its pro-$\mathbf{V}$ topology. Our main result states that it is decidable whether $H$ is $\mathbf{Su}$-dense, where $\mathbf{Su}\subset \mathbf{S}$ denote respectively the pseudovarieties of all finite supersolvable groups and all finite solvable groups. Our motivation stems from the following open problem: is it decidable whether $H$ is $\mathbf{S}$-dense?

math.GR↗

arXiv2023

The pro-$k$-solvable topology on a free group

We prove that, given a finitely generated subgroup $H$ of a free group $F$, the following questions are decidable: is $H$ closed (dense) in $F$ for the pro-(met)abelian topology? is the closure of $H$ in $F$ for the pro-(met)abelian topology finitely generated? We show also that if the latter question has a positive answer, then we can effectively construct a basis for the closure, and the closure has decidable membership problem in any case. Moreover, it is decidable whether $H$ is closed for the pro-${\bf V}$ topology when ${\bf V}$ is an equational pseudovariety of finite groups, such as the pseudovariety ${\bf S}_k$ of all finite solvable groups with derived length $\leq k$. We also connect the pro-abelian topology with the topologies defined by abelian groups of bounded exponent.

math.GR↗

arXiv2018

Generating maximal subgroups of finite almost simple groups

For a finite group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. In this paper, given a finite almost simple group $G$ and any maximal subgroup $H$ of $G$, we determine a precise upper bound for $d(H)$. In particular, we show that $d(H)\leq 5$, and that $d(H)\geq 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.

math.GR↗

arXiv2017

On finite simple images of triangle groups

For a simple algebraic group G in characteristic p, a triple (a,b,c) of positive integers is said to be rigid for G if the dimensions of the subvarieties of G of elements of order dividing a,b,c sum to 2dim G. In this paper we complete the proof of a conjecture of the third author, that for a rigid triple (a,b,c) for G with p>0, the triangle group T_{a,b,c} has only finitely many simple images of the form G(p^r). We also obtain further results on the more general form of the conjecture, where the images G(p^r) can be arbitrary quasisimple groups of type G.

math.GR↗

arXiv2017

Varieties of elements of given order in simple algebraic groups

Given a positive integer $u$ and a simple algebraic group $G$ defined over an algebraically closed field $K$ of characteristic $p$, we derive properties about the subvariety $G_{[u]}$ of $G$ consisting of elements of $G$ of order dividing $u$. In particular, we determine the dimension of $G_{[u]}$, completing results of Lawther [7] in the special case where $G$ is of adjoint type. We also apply our results to the study of finite simple quotients of triangle groups, giving further insight on a conjecture we proposed in [10] as well as proving that some finite quasisimple groups are not quotients of certain triangle groups.

math.GR↗

arXiv2017

Alternating and symmetric groups with Eulerian generating graph

Given a finite group $G$, the generating graph $Γ(G)$ of $G$ has as vertices the (nontrivial) elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. In this paper we investigate properties about the degrees of the vertices of $Γ(G)$ when $G$ is an alternating group or a symmetric group. In particular, we determine the vertices of $Γ(G)$ having even degree and show that $Γ(G)$ is Eulerian if and only if $n$ and $n-1$ are not equal to a prime number congruent to 3 modulo 4.

math.GR↗

arXiv2016

On irreducible subgroups of simple algebraic groups

Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p\geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial irreducible $KG$-module, which is $p$-restricted, tensor indecomposable and rational. Assume that the restriction of $V$ to $H$ is irreducible. In this paper, we study the triples $(G,H,V)$ of this form when $G$ is a classical group and $H$ is positive-dimensional. Combined with earlier work of Dynkin, Seitz, Testerman and others, our main theorem reduces the problem of classifying the triples $(G,H,V)$ to the case where $G$ is an orthogonal group, $V$ is a spin module and $H$ normalizes an orthogonal decomposition of the natural $KG$-module.

math.GR↗

arXiv2013

Irreducible almost simple subgroups of classical algebraic groups

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic $p \ge 0$ with natural module W. Let H be a closed subgroup of G and let V be a nontrivial p-restricted irreducible tensor indecomposable rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V) of this form, where $V \ne W, W^{*}$ and H is a disconnected almost simple positive-dimensional closed subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classification of the irreducible triples (G,H,V) where G is a simple algebraic group over K, and H is a maximal closed subgroup of positive dimension.

math.GR↗

arXiv2013

Deformation theory and finite simple quotients of triangle groups I

Let $2 \leq a \leq b \leq c \in \mathbb{N}$ with $μ=1/a+1/b+1/c<1$ and let $T=T_{a,b,c}=< x,y,z: x^a=y^b=z^c=xyz=1>$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$? (Classically, for $(a,b,c)=(2,3,7)$ and more recently also for general $(a,b,c)$.) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially prove a conjecture of Marion [21] showing that various finite simple groups are not quotients of $T$, as well as positive results showing that many finite simple groups are quotients of $T$.

math.GR↗

arXiv2013

Deformation theory and finite simple quotients of triangle groups II

This paper is a continuation of our first paper [10] in which we showed how deformation theory of representation varieties can be used to study finite simple quotients of triangle groups. While in Part I, we mainly used deformations of the principal homomorphism from ${\rm SO}(3,\R)$, in this part we use ${\rm PGL}_2(\R)$ as well as deformations of representations which are very different from the principal homomorphism.

math.GR↗