arXiv subjects

D. Bubboloni

Publications and source records attributed to D. Bubboloni.

3 recordsLinked to original sources

arXiv2017

Quotient graphs for power graphs

In a previous paper of the first author a procedure was developed for counting the components of a graph through the knowledge of the components of its quotient graphs. We apply here that procedure to the proper power graph $\mathcal{P}_0(G)$ of a finite group $G$, finding a formula for the number $c(\mathcal{P}_0(G))$ of its components which is particularly illuminative when $G\leq S_n$ is a fusion controlled permutation group. We make use of the proper quotient power graph $\widetilde{\mathcal{P}}_0(G)$, the proper order graph $\mathcal{O}_0(G)$ and the proper type graph $\mathcal{T}_0(G)$. We show that all those graphs are quotient of $\mathcal{P}_0(G)$ and demonstrate a strong link between them dealing with $G=S_n$. We find simultaneously $c(\mathcal{P}_0(S_n))$ as well as the number of components of $\widetilde{\mathcal{P}}_0(S_n)$, $\mathcal{O}_0(S_n)$ and $\mathcal{T}_0(S_n)$.

math.CO↗

arXiv2015

Intersective $S_n$ polynomials with few irreducible factors

An intersective polynomial is a monic polynomial in one variable with rational integer coefficients, with no rational root and having a root modulo $m$ for all positive integers $m$. Let $G$ be a finite noncyclic group and let $r(G)$ be the smallest number of irreducible factors of an intersective polynomial with Galois group $G$ over $\mathbb{Q}$. Let $s(G)$ be smallest number of proper subgroups of $G$ having the property that the union of their conjugates is $G$ and the intersection of all their conjugates is trivial. It is known that $s(G)\leq r(G).$ It is also known that if $G$ is realizable as a Galois group over the rationals, then it is also realizable as the Galois group of an intersective polynomial. However it is not known, in general, whether there exists such a polynomial which is a product of the smallest feasible number $s(G)$ of irreducible factors. In this paper, we study the case $G=S_n$, the symmetric group on $n$ letters. We prove that for every $n$, either $r(S_n)=s(S_n)$ or $r(S_n)=s(S_n)+1$ and that the optimal value $s(S_n)$ is indeed attained for all odd $n$ and for some even $n$. Moreover, we compute $r(S_n)$ when $n$ is the product of at most two odd primes and we give general upper and lower bounds for $r(S_n).$

math.GR↗

arXiv2011

2-Coverings of classical groups

In this paper we show that if $n\geq 5$ and $G$ is any of the groups $SU_n(q)$ with $n\neq 6,$ $Sp_{2n}(q)$ with $q$ odd, $Ω_{2n+1}(q),$ $Ω_{2n}^{\pm}(q),$ then $G$ and the simple group $\barG=G/Z(G)$ are not 2-coverable. Moreover the only 2-covering of $Sp_{2n}(q),$ with $q$ even, has components $ O^-_{2n}(q)$ and $O^{+}_{2n}(q) .$

math.GR↗