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Martino Garonzi

Publications and source records attributed to Martino Garonzi.

33 records · Page 2Linked to original sources
arXiv2018

On the number of cyclic subgroups of a finite group

Let $G$ be a finite group and let $c(G)$ be the number of cyclic subgroups of $G$. We study the function $α(G) = c(G)/|G|$. We explore its basic properties and we point out a connection with the probability of commutation. For many families $\mathscr{F}$ of groups we characterize the groups $G \in \mathscr{F}$ for which $α(G)$ is maximal and we classify the groups $G$ for which $α(G) > 3/4$. We also study the number of cyclic subgroups of a direct power of a given group deducing an asymptotic result and we characterize the equality $α(G) = α(G/N)$ when $G/N$ is a symmetric group.

math.GR↗
arXiv2017

Finite Groups with 6 or 7 Automorphism Orbits

Let $G$ be a group. The orbits of the natural action of $\mbox{Aut}(G)$ on $G$ are called "automorphism orbits" of $G$, and the number of automorphism orbits of $G$ is denoted by $ω(G)$. In this paper the finite nonsolvable groups $G$ with $ω(G) \leq 6$ are classified - this solves a problem posed by Markus Stroppel - and it is proved that there are infinitely many finite nonsolvable groups $G$ with $ω(G)=7$. Moreover it is proved that for a given number $n$ there are only finitely many finite groups $G$ without nontrivial abelian normal subgroups and such that $ω(G) \leq n$, generalizing a result of Kohl.

math.GR↗
arXiv2015

Inequalities detecting structural properties of a finite group

We prove several results detecting ciclicity or nilpotency of a finite group $G$ in terms of inequalities involving the orders of the elements of $G$ and the orders of the elements of the cyclic group of order $|G|$. We prove that, among the groups of the same order, the number of cyclic subgroups is minimal for the cyclic group and the product of the orders of the elements is maximal for the cyclic group.

math.GR↗
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↗
arXiv2014

Irredundant and minimal covers of finite groups

A cover of a finite non-cyclic group $G$ is a family $\mathcal{H}$ of proper subgroups of $G$ whose union equals $G$. A cover of $G$ is called minimal if it has minimal size, and irredundant if it does not properly contain any other cover. We classify the finite non-cyclic groups all of whose irredundant covers are minimal.

math.GR↗
arXiv2014

Factorizing a Finite Group into Conjugates of a Subgroup

For every non-nilpotent finite group $G$, there exists at least one proper subgroup $M$ such that $G$ is the setwise product of a finite number of conjugates of $M$. We define $γ_{\text{cp}}\left( G\right) $ to be the smallest number $k$ such that $G$ is a product, in some order, of $k$ pairwise conjugated proper subgroups of $G$. We prove that if $G$ is non-solvable then $γ_{\text{cp}}\left( G\right) \leq36$ while if $G$ is solvable then $γ_{\text{cp}}\left( G\right) $ can attain any integer value bigger than $2$, while, on the other hand, $γ_{\text{cp}}\left( G\right) \leq4\log_{2}\left\vert G\right\vert $.

math.GR↗
arXiv2013

Covers and Normal Covers of Finite Groups

For a finite non cyclic group $G$, let $γ(G)$ be the smallest integer $k$ such that $G$ contains $k$ proper subgroups $H_1,\dots,H_k$ with the property that every element of $G$ is contained in $H_i^g$ for some $i \in \{1,\dots,k\}$ and $g \in G.$ We prove that if $G$ is a noncyclic permutation group of degree $n,$ then $γ(G)\leq (n+2)/2.$ We then investigate the structure of the groups $G$ with $γ(G)=σ(G)$ (where $σ(G)$ is the size of a minimal cover of $G$) and of those with $γ(G)=2.$

math.GR↗
arXiv2013

Covering monolithic groups with proper subgroups

Given a finite non-cyclic group $G$, call $σ(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $σ(G) < σ(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is \textit{monolithic}, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.

math.GR↗
arXiv2012

Covering certain Wreath Products with Proper Subgroups

For a non-cyclic finite group $X$ let $σ(X)$ be the least number of proper subgroups of $X$ whose union is $X$. Precise formulas or estimates are given for $σ(S \wr C_{m})$ for certain nonabelian finite simple groups $S$ where $C_m$ is a cyclic group of order $m$.

math.GR↗
arXiv2012

Covering certain monolithic groups with proper subgroups

Given a finite non-cyclic group $G$, call $σ(G)$ the least number of proper subgroups of $G$ needed to cover $G$. In this paper we give lower and upper bounds for $σ(G)$ for $G$ a group with a unique minimal normal subgroup $N$ isomorphic to $A_n^m$ where $n \geq 5$ and $G/N$ is cyclic. We also show that $σ(A_5 \wr C_2)=57$.

math.GR↗
arXiv2011

Finite Groups that are the union of at most 25 proper subgroups

For a finite group $G$ let $σ(G)$ (the "sum" of $G$) be the least number of proper subgroups of $G$ whose set-theoretical union is equal to $G$, and $σ(G)=\infty$ if $G$ is cyclic. We say that a group $G$ is $σ$-elementary if for every non-trivial normal subgroup $N$ of $G$ we have $σ(G)<σ(G/N)$. In this paper we produce the list of all the $σ$-elementary groups of sum up to 25. We also show that $σ(\Aut(PSL(2,8)))=29$.

math.GR↗