Searcharxiv⌕ Search

arXiv subjects

Patrizia Longobardi

Publications and source records attributed to Patrizia Longobardi.

10 recordsLinked to original sources
arXiv2024

Limit Groups and Automorphisms of $κ$-Existentially Closed Groups

The structure of automorphism groups of $κ$-existentially closed groups are studied by Kaya-Kuzucuoğlu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving automorphisms and $|Aut(G)|=2^κ$ whenever $κ$ is an inaccessible cardinal and $G$ is the unique $κ$-existentially closed group of cardinality $κ$. The cardinality of the automorphism group of a $κ$-existentially closed group of cardinality $λ>κ$ is asked in Kourovka Notebook Question 20.40. Here we answer positively the promised case $κ=λ$ namely: If $G$ is a $κ$-existentially closed group of cardinality $κ$, then $|Aut(G)|=2^κ$. We also answer Kegel's question on universal groups, namely: For any uncountable cardinal $κ$, there exist universal groups of cardinality $κ$.

math.LO↗
arXiv2023

On conjygacy classes in groups

Let $G$ be a group. Write $G^{*}=G\setminus \{1\}$. An element $x$ of $G^{*}$ will be called deficient if $ \langle x\rangle < C_G(x)$ and it will be called non-deficient if $\langle x\rangle = C_G(x).$ If $x\in G$ is deficient (non-deficient), then the conjugacy class $x^G$ of $x$ in $G$ will be also called deficient (non-deficient). Let $j$ be a non-negative integer. We shall say that the group $G$ has defect $j$, denoted by $G\in D(j)$ or by the phrase "$G$ is a $D(j)$-group", if exactly $j$ non-trivial conjugacy classes of $G$ are deficient. We first determine all finite $D(0)$-groups and $D(1)$-groups. Then we deal with arbitrary $D(0)$-groups and $D(1)$-groups: we find properties of arbitrary $D(0)$-groups and $D(1)$-groups, which force these groups to be finite.

math.GR↗
arXiv2022

On groups in which every element has a prime power order and which satisfy some boundedness condition

In this paper we shall deal with periodic groups, in which each element has a prime power order. A group $G$ will be called a $BCP$-group if each element of $G$ has a prime power order and for each $p\in π(G)$ there exists a positive integer $u_p$ such that each $p$-element of $G$ is of order $p^i\leq p^{u_p}$. A group $G$ will be called a $BSP$-group if each element of $G$ has a prime power order and for each $p\in π(G)$ there exists a positive integer $v_p$ such that each finite $p$-subgroup of $G$ is of order $p^j\leq p^{v_p}$. Here $π(G)$ denotes the set of all primes dividing the order of some element of $G$. Our main results are the following four theorems. Theorem 1: Let $G$ be a finitely generated $BCP$-group. Then $G$ has only a finite number of normal subgroups of finite index. Theorem 4: Let $G$ be a locally graded $BCP$-group. Then $G$ is a locally finite group. Theorem 7: Let $G$ be a locally graded $BSP$-group. Then $G$ is a finite group. Theorem 9: Let $G$ be a $BSP$-group satisfying $2\in π(G)$. Then $G$ is a locally finite group.

math.GR↗
arXiv2020

On the structure of some locally nilpotent groups without contranormal subgroups

Following J.S. Rose, a subgroup H of a group G is said contranormal in G if G = H^G . In a certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. We prove that a nilpotent-by-finite group with no proper contranormal subgroup is nilpotent. There are locally nilpotent groups with a proper contranormal subgroup. We study the structure of hypercentral groups with a finite proper contranormal subgroup.

math.GR↗
arXiv2019

Sums of element orders in groups of odd order

Denote by $G$ a finite group and by $ψ(G)$ the sum of element orders in $G$. If $t$ is a positive integer, denote by $C_t$ the cyclic group of order $t$ and write $ψ(t)=ψ(C_t)$. In this paper we proved the following Theorem A: Let $G$ be a non-cyclic group of odd order $n=qm$, where $q$ is the smallest prime divisor of $n$ and $(m,q)=1$. Then the following statements hold. (1) If $q=3$, then $\frac {ψ(G)}{ψ(|G|)}\leq \frac {85}{301}$, and equality holds if and only if $n=3\cdot 7\cdot m_1$ with $(m_1,42)=1$ and $G=(C_7\rtimes C_3)\times C_{m_1}$, with $C_7\rtimes C_3$ non-abelian. (2) If $q>3$, then $\frac {ψ(G)}{ψ(|G|)}\leq \frac {p^4+p^3-p^2+1}{p^5+1}$, where $p$ is the smallest prime bigger than $q$ and equality holds if and only if $n=qp^2m_1$ with $(m_1,p!)=1$ and $G=C_q\times C_p\times C_p \times C_{m_1}$.

math.GR↗
arXiv2019

The second maximal groups with respect to the sum of element orders

Denote by $G$ a finite group and let $ψ(G)$ denote the sum of element orders in $G$. In 2009, H.Amiri, S.M.Jafarian Amiri and I.M.Isaacs proved that if $|G|=n$ and $G$ is non-cyclic, then $ψ(G)<ψ(C_n)$, where $C_n$ denotes the cyclic group of order $n$. In 2018 we proved that if $G$ is non-cyclic group of order $n$, then $ψ(G)\leq \frac 7{11}ψ(C_n)$ and equality holds if $n=4k$ with $(k,2)=1$ and $G=(C_2\times C_2)\times C_k$. In this paper we proved that equality holds if and only if $n$ and $G$ are as indicated above. Moreover we proved the following generalization of this result: Theorem 4. Let $q$ be a prime and let $G$ be a non-cyclic group of order $n$, with $q$ being the least prime divisor of $n$. Then $ψ(G)\leq \frac {((q^2-1)q+1)(q+1)}{q^5+1}ψ(C_n)$, with equality if and only if $n=q^2k$ with $(k,q)=1$ and $G=(C_q\times C_q)\times C_k$. Notice that if $q=2$, then $\frac {((q^2-1)q+1)(q+1)}{q^5+1}=\frac 7{11}$.

math.GR↗
arXiv2016

An exact upper bound for sums of element orders in non-cyclic finite groups

Denote the sum of element orders in a finite group $G$ by $ψ(G)$ and let $C_n$ denote the cyclic group of order $n$. Suppose that $G$ is a non-cyclic finite group of order $n$ and $q$ is the least prime divisor of $n$. We proved that $ψ(G)\leq\frac 7{11}ψ(C_n)$ and $ψ(G)<\frac 1{q-1}ψ(C_n)$. The first result is best possible, since for each $n=4k$, $k$ odd, there exists a group $G$ of order $n$ satisfying $ψ(G)=\frac 7{11}ψ(C_n)$ and the second result implies that if $G$ is of odd order, then $ψ(G)<\frac 12ψ(C_n)$. Our results improve the inequality $ψ(G)<ψ(C_n)$ obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some $ψ(G)$-based sufficient conditions for the solvability of $G$.

math.GR↗
arXiv2015

Small doubling in ordered groups: generators and structure

We prove several new results on the structure of the subgroup generated by a small doubling subset of an ordered group, abelian or not. We obtain precise results generalizing Freiman's 3k-3 and 3k-2 theorems in the integers and several further generalizations.

math.NT↗