A result on the Chermak-Delgado measure of a finite group
In this short note, we describe finite groups all of whose non-trivial cyclic subgroups have the same Chermak-Delgado measure.
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arXiv subjects
Marius Tărnăuceanu
Publications and source records attributed to Marius Tărnăuceanu.
On CLT and non-CLT groups
In this note, we prove that for every integer $d\geq 2$ which is not a prime power, there exists a finite solvable group $G$ such that $d\mid |G|$, $π(G)=π(d)$ and $G$ has no subgroup of order $d$. We also introduce the CLT-degree of a finite group and answer two questions about it.
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On groups occurring as absolute centers of finite groups
Given a construction $f$ on groups, we say that a group $G$ is \textit{$f$-realisable} if there is a group $H$ such that $G\cong f(H)$, and \textit{completely $f$-realisable} if there is a group $H$ such that $G\cong f(H)$ and every subgroup of $G$ is isomorphic to $f(H_1)$ for some subgroup $H_1$ of $H$ and vice versa. Denote by $L(G)$ the absolute center of a group $G$, that is the set of elements of $G$ fixed by all automorphisms of $G$. By using the structure of the automorphism group of a ZM-group, in this paper we prove that cyclic groups $C_N$, $N\in\mathbb{N}^*$, are completely $L$-realisable.
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Finite groups with integer harmonic mean of element orders
In this paper, we introduce a new function computing the harmonic mean of element orders of a finite group. We present a series of properties for this function, and then we study groups for which the value of the function is an integer.
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Completely realisable groups
Given a construction $f$ on groups, we say that a group $G$ is \textit{$f$-realisable} if there is a group $H$ such that $G\cong f(H)$, and \textit{completely $f$-realisable} if there is a group $H$ such that $G\cong f(H)$ and every subgroup of $G$ is isomorphic to $f(H_1)$ for some subgroup $H_1$ of $H$ and vice versa. In this paper, we determine completely ${\rm Aut}$-realisable groups. We also study $f$-realisable groups for $f=Z,F,M,D,Φ$, where $Z(H)$, $F(H)$, $M(H)$, $D(H)$ and $Φ(H)$ denote the center, the Fitting subgroup, the Chermak-Delgado subgroup, the derived subgroup and the Frattini subgroup of the group $H$, respectively.
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A characterization of $A_5$ by its average order
Let $o(G)$ be the average order of a finite group $G$. M. Herzog, P. Longobardi and M. Maj \cite{7} showed that if $G$ is non-solvable and $o(G)=o(A_5)$, then $G\cong A_5$. In this note, we prove that the equality $o(G)=o(A_5)$ does not hold for any finite solvable group $G$. Consequently, up to isomorphism, $A_5$ is determined by its average order.
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On the number of edges of cyclic subgroup graphs of finite groups
In this note, we show that among finite nilpotent groups of a given order or finite groups of a given odd order, the cyclic group of that order has the minimum number of edges in its cyclic subgroup graph. We also conjecture that this holds for arbitrary finite groups.
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On a group-theoretical generalization of the Gauss formula
In this paper, we discuss a group-theoretical generalization of the well-known Gauss formula involving the functionthat counts the number of automorphisms of a finite group. This gives several characterizations of finite cyclic groups.
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On the average order of a finite group
Let $o(G)$ be the average order of a finite group $G$. We show that if $o(G)<c$, where $c\in \lbrace \frac{13}{6}, \frac{11}{4}\rbrace$, then $G$ is an elementary abelian 2-group or a solvable group, respectively. Also, we prove that the set containing the average orders of all finite groups is not dense in $[a, \infty)$, for all $a\in [0, \frac{13}{6}]$. We also outline some results related to the integer values of the average order. Since group element orders is a popular research topic, we pose some open problems concerning the average order of a finite group throughout the paper.
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Finite groups with large Chermak-Delgado lattices
Given a finite group $G$, we denote by $L(G)$ the subgroup lattice of $G$ and by ${\cal CD}(G)$ the Chermak-Delgado lattice of $G$. In this note, we determine the finite groups $G$ such that $|{\cal CD}(G)|=|L(G)|-k$, $k=1,2$.
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Another criterion for supersolvability of finite groups
Let $o(G)$ be the average order of a finite group $G$. In this paper, we prove that if $o(G)<\frac{31}{12}$\,, then $G$ is supersolvable. Moreover, we have $o(G)=\frac{31}{12}$ if and only if $G\cong A_4$. We also classify finite groups $G$ satisfying $o(G)<\frac{31}{12}$\,.
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Addendum to "A generalization of a result on the sum of element orders of a finite group" [arXiv:2001.07275]
Let $G$ be a group of order $n$ and $H$ be a subgroup of order $m$ of $G$. Denote by $ψ_H(G)$ the sum of element orders relative to $H$ of $G$. It is known that if $G$ is nilpotent, then $ψ_H(G)\leqψ_{H_m}(G)$, where $H_m$ is the unique subgroup of order $m$ of $C_n$. In this note, we show that this inequality does not hold for infinitely many finite solvable groups.
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Finite groups with many isolated subgroups
Given a finite group $G$, we denote by $L(G)$ the subgroup lattice of $G$ and by ${\rm Isolated}(G)$ the set of isolated subgroups of $G$. In this note, we describe finite groups $G$ such that $|{\rm Isolated}(G)|=|L(G)|-k$, where $k=0,1,2$.
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Generalizing the Lehmer's totient problem
An important unsolved question in number theory is the Lehmer's totient problem that asks whether there exists any composite number $n$ such that $φ(n)\mid n-1$, where $φ$ is the Euler's totient function. It is known that if any such $n$ exists, it must be odd, square-free, greater that $10^{30}$, and divisible by at least $15$ distinct primes. Such a number must be also a Carmichael number. In this short note, we discuss a group-theoretical analogous problem involving the function that counts the number of automorphisms of a finite group. Another way to generalize the Lehmer's totient problem is also proposed.
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On a group-theoretical generalization of the Euler's totient function
Let $G$ be a finite group and $φ(G)=|\{a\in G \mid o(a)=\exp(G)\}|$, where $o(a)$ denotes the order of $a$ in $G$ and $\exp(G)$ denotes the exponent of $G$. Under a natural hypothesis, in this note we determine the groups $G$ such that $\forall\, H,K\leq G$, $H\subseteq K$ implies $φ(H)\midφ(K)$. This partially answers Problem 5.4 in \cite{6}.
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A new Menon-type identity derived from group actions
In this short note, we give a new Menon-type identity involving the sum of element orders and the sum of cyclic subgroup orders of a finite group. It is based on applying the weighted form of Burnside's lemma to a natural group action.
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On the supersolvability of a finite group by the sum of subgroup orders
Let $G$ be a finite group and $σ_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this paper, we prove that if $σ_1(G)<2+\frac{11}{|G|}$\,, then $G$ is supersolvable. In particular, some new characterizations of the well-known groups $\mathbb{Z}_2\times\mathbb{Z}_4$ and $A_4$ are obtained. We also show that $σ_1(G)<c$ does not imply the supersolvability of $G$ for no constant $c\in(2,\infty)$.
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Isolated subgroups of finite abelian groups
We say that a subgroup $H$ is isolated in a group $G$ if for every $x\in G$ we have either $x\in H$ or $\langle x\rangle\cap H=1$. In this short note, we describe the set of isolated subgroups of a finite abelian group. The technique used is based on an interesting connection between isolated subgroups and the function sum of element orders of a finite group.
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