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Pietro Gheri

Publications and source records attributed to Pietro Gheri.

4 recordsLinked to original sources
arXiv2022

On the maximal number of elements pairwise generating the finite alternating group

Let $G$ be the alternating group of degree $n$. Let $ω(G)$ be the maximal size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y \in S$ and $x \neq y$ and let $σ(G)$ be the minimal size of a family of proper subgroups of $G$ whose union is $G$. We prove that, when $n$ varies in the family of composite numbers, $σ(G)/ω(G)$ tends to $1$ as $n \to \infty$. Moreover, we explicitly calculate $σ(A_n)$ for $n \geq 21$ congruent to $3$ modulo $18$.

math.GR↗
arXiv2020

Subnormalizers and solvability in finite groups

For a finite group $G$, we study the probability $sp(G)$ that, given two elements $x,y \in G$, the cyclic subgroup $\langle x \rangle$ is subnormal in the subgroup $\langle x, y \rangle$. This can be seen as an intermediate invariant between the probability that two elements generate a nilpotent subgroup and the probability that two elements generate a solvable subgroup. We prove that $sp(G) \leq 1/6$ for every nonsolvable group $G$.

math.GR↗
arXiv2020

On the number of $p$-elements in a finite group

In this paper we study the ratio between the number of $p$-elements and the order of a Sylow $p$-subgroup of a finite group $G$. As well known, this ratio is a positive integer and we conjecture that, for every group $G$, it is at least the $(1-\frac{1}{p})$-th power of the number of Sylow $p$-subgroups of $G$. We prove this conjecture if $G$ is $p$-solvable. Moreover, we prove that the conjecture is true in its generality if a somewhat similar condition holds for every almost simple group.

math.GR↗