Finite groups with small number of cyclic subgroups

Finite groups with a certain number of cyclic subgroups II

In this note we describe the finite groups $G$ having $|G|-2$ cyclic subgroups. This partially solves the open problem in the end of \cite{3}.

Locally finite groups in which every non-cyclic subgroup is self-centralizing

Locally finite groups having the property that every non-cyclic subgroup contains its centralizer are completely classified.

Essential Cohomology of the p-Groups with a Cyclic Subgroup of Index p

In this paper we determine explicitly the mod-p essential cohomology ideals of the p-groups with a cyclic subgroup of index p.

On finite capable $p$-groups of class 2 with cyclic commutator subgroups

We study finite capable $p$-groups $G$ of nilpotency class 2 such that the commutator subgroup $γ_2(G)$ of $G$ is cyclic and the center of $G$ is contained in the Frattini subgroup of $G$.

Separating cyclic subgroups in graph products of groups

We prove that the property of being cyclic subgroup separable, that is having all cyclic subgroups closed in the profinite topology, is preserved under forming graph products. Furthermore, we develop the tools to study the analogous question in the pro-$p$ case. For a wide class of groups we show that the relevant cyclic subgroups - which are called $p$-isolated - are closed in the pro-$p$ topology of the graph product. In particular, we show that every $p$-isolated cyclic subgroup of a right-angled Artin group is closed in the pro-$p$ topology, and we fully characterise such subgroups.

Cyclic subgroup commutativity degrees of finite groups

In this paper we introduce and study the concept of cyclic subgroup commutativity degree of a finite group $G$. This quantity measures the probability of two random cyclic subgroups of $G$ commuting. Explicit formulas are obtained for some particular classes of groups. A criterion for a finite group to be an Iwasawa group is also presented.

Relative cyclic subgroup commutativity degrees of finite groups

In this paper we introduce and study the relative cyclic subgroup commutativity degrees of a finite group. We show that there is a finite group with $n$ such degrees for all $n \in \mathbb{N}^*\setminus \lbrace 2\rbrace$ and we indicate some classes of finite groups with few relative cyclic subgroup commutativity degrees. Using this new concept, we are able to show that the set containing all cyclic subgroup commutativity degrees of finite groups is not dense in $[0, 1]$.

Permutability graph of cyclic subgroups

Let $G$ be a group. \textit{The permutability graph of cyclic subgroups of $G$}, denoted by $Γ_c(G)$, is a graph with all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $Γ_c(G)$ are adjacent if and only if the corresponding subgroups permute in $G$. In this paper, we classify the finite groups whose permutability graph of cyclic subgroups belongs to one of the following: bipartite, tree, star graph, triangle-free, complete bipartite, $P_n$, $C_n$, $K_4$, $K_{1,3}$-free, unicyclic. We classify abelian groups whose permutability graph of cyclic subgroups are plan…

On the cyclic subgroup separability of the free product of two groups with commuting subgroups

Let G be the free product of groups A and B with commuting subgroups H \leqslant A and K \leqslant B, and let C be the class of all finite groups or the class of all finite p-groups. We derive the description of all C-separable cyclic subgroups of G provided this group is residually a C-group. We prove, in particular, that if A, B are finitely generated nilpotent groups and H, K are p'-isolated in the free factors, then all p'-isolated cyclic subgroups of G are separable in the class of all finite p-groups. The same statement is true provided A, B are free and H, K are p'-isolated and cyclic.

On the cyclic subgroup separability of free products of two groups with amalgamated subgroup

Let $G$ be a free product of two groups with amalgamated subgroup, $π$ be either the set of all prime numbers or the one-element set \{$p$\} for some prime number $p$. Denote by $Σ$ the family of all cyclic subgroups of group $G$, which are separable in the class of all finite $π$-groups. Obviously, cyclic subgroups of the free factors, which aren't separable in these factors by the family of all normal subgroups of finite $π$-index of group $G$, the subgroups conjugated with them and all subgroups, which aren't $π^{\prime}$-isolated, don't belong to $Σ$. Some sufficient conditions are obtaine…

Intersections of Magnus subgroups and embedding theorems for cyclically presented groups

A Magnus subgroup of a one-relator group is the free subgroup freely generated by a proper subset of the generators. Two such subgroups can intersect in the obvious way or in a larger, exceptional way. The condition of non-exceptional intersection of Magnus subgroups of a one-relator group is applied to give criteria for the resulting cyclically presented groups to contain nonabelian free subgroups.

Generation of finite groups with cyclic Sylow subgroups

Slattery (2007) described computational methods to enumerate, construct, and identify finite groups of squarefree order. We generalise Slattery's result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP.

On the number of cyclic transitive subgroups of a permutation group

We prove an upper bound for the number of cyclic transitive subgroups in a finite permutation group and clarify the structure of the groups for which this bound becomes sharp. We also give an application in the theory of number fields.

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.

Belk-Hyde-Matucci群$V\!\mathcal{A}$的循环子群

Cyclic Subgroups of Belk-Hyde-Matucci Group $V\!\mathcal{A}$

本文证明Belk-Hyde-Matucci群$V\!\mathcal{A}$不包含具有扭曲循环子群的子群。

The number of cyclic subgroups of finite abelian groups and Menon's identity

In this note, we give a new formula for the number of cyclic subgroups of a finite abelian group. This is based on applying the Burnside's lemma to a certain group action. Also, it generalizes the well-known Menon's identity.

Finite groups with $\mathbb P$-subnormal primary cyclic subgroups

A subgroup $H$ of a group $G$ is called $\mathbb P$-subnormal in $G$ whenever either $H=G$ or there is a chain of subgroups $H=H_0\subset H_1\subset ... \subset H_n=G$ such that $|H_i:H_{i-1}|$ is a prime for all $i$. In this paper, we study the groups in which all primary cyclic subgroups are $\mathbb P$-subnormal.

Groups in which every non-cyclic subgroup contains its centralizer

We study groups having the property that every non-cyclic subgroup contains its centralizer. The structure of nilpotent and supersolvable groups in this class is described. We also classify finite $p$-groups and finite simple groups with the above defined property.

Amenable actions of amalgamated free products of free groups over a cyclic subgroup and generic property

We show that the class of amalgamated free products of two free groups over a cyclic subgroup admits amenable, faithful and transitive actions on infinite countable sets. This work generalizes the results on such actions for doubles of free group on 2 generators.

有限群中每个非平凡特征子群都是循环群的群

Finite groups in which every proper characteristic subgroup is cyclic

本文研究了所有非平凡特征子群均为循环群的有限群，对其进行了完整分类，并推动了最小非循环斜 brace 的分类研究。

Limit groups over coherent right-angled Artin groups are cyclic subgroup separable

We prove that cyclic subgroup separability is preserved under exponential completion for groups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolic groups; we do so by exploiting the structure of these completions as iterated free products with commuting subgroups. From this we deduce that the cyclic subgroups of limit groups over coherent RAAGs are separable, answering a question of Casals-Ruiz, Duncan and Kazachov. We also discuss relations between free products with commuting subgroups and the word problem, and recover the fact that limit groups over cohe…

Finite metacyclic groups as active sums of cyclic subgroups

The notion of active sum provides an analogue for groups of that of direct sum for abelian groups. One natural question then is which groups are the active sum of cyclic subgroups. Many groups have been found to give a positive answer to this question, while the case of finite metacyclic groups remained unknown. In this note we show that every finite metacyclic group can be recovered as the active sum of a discrete family of cyclic subgroups.

Hyperbolic modules and cyclic subgroups

Let $G$ be a finite group of odd order, $\F$ a finite field of odd characteristic $p$ and $\B$ a finite--dimensional symplectic $\F G$-module. We show that $\B$ is $\F G$-hyperbolic, i.e., it contains a self--perpendicular $\F G$-submodule, iff it is $\F N$-hyperbolic for every cyclic subgroup $N$ of $G$.

A note on cyclic semiregular subgroups of some 2-transitive permutation groups

We determine the semi-regular subgroups of the 2-transitive permutation groups PGL(2,n), PSL(2,n), PGU(3,n), PSU(3,n), Sz(n) and Ree(n) with n a suitable power of a prime number p.

Finite non-cyclic $p$-groups whose number of subgroups is minimal

Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groups which are not elementary abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results from the bottom level up by determining completely the non-cyclic finite p-groups whose number of subgroups among p-groups of a given order is minimal.

Conjugacy classes of maximal cyclic subgroups and nilpotence class of $p$-groups

In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We prove that if $G$ is a $p$-group of order $p^n$ and nilpotence class $l$, then $η(G)$ is bounded below by a linear function in $n/l$.

Finite groups with specific number of cyclic subgroups

In this note, we classify all finite groups having exactly 6, 7 or 8 cyclic subgroups. This gives a partial answer to the open problem posed by Tarnauceanu (Amer. Math. Monthly, 122 (2015), 275-276). As a consequence of our results, we also obtain an important result concerning with the converse of Lagrange's theorem.

A result on the number of cyclic subgroups of a finite group

Let $G$ be a finite group and $α(G)=\frac{|C(G)|}{|G|}$\,, where $C(G)$ denotes the set of cyclic subgroups of $G$. In this short note, we prove that $α(G)\leqα(Z(G))$ and we describe the groups $G$ for which the equality occurs. This gives some sufficient conditions for a finite group to be $4$-abelian or abelian.

Intersection graph of cyclic subgroups of groups

Let $G$ be a group. The intersection graph of cyclic subgroups of $G$, denoted by $\mathscr I_c(G)$, is a graph having all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\mathscr I_c(G)$ are adjacent if and only if their intersection is non-trivial. In this paper, we classify the finite groups whose intersection graph of cyclic subgroups is one of totally disconnected, complete, star, path, cycle. We show that for a given finite group $G$, $girth(\mathscr I_c (G)) \in \{3, \infty\}$. Moreover, we classify all finite non-cyclic abelian groups whose intersection…

The cyclic subgroup separability of certain generalized free products of two groups

Free products of two residually finite groups with amalgamated retracts are considered. It is proved that a cyclic subgroup of such a group is not finitely separable if, and only if, it is conjugated with a subgroup of a free factor which is not finitely separable in this factor. A similar result is obtained for the case of separability in the class of finite p-groups.

按循环子群数量给出的界

Bounds in terms of the number of cyclic subgroups

该研究提出（极大）循环有界群族的概念，证明素数幂阶群族为极大循环有界、无互素循环直因子的有限群族为循环有界，并给出判定不小于10的自然数为素数的等价条件。

A note on groups with self-normalizing cyclic subgroups

In this article, we prove that if all non-trivial cyclic subgroups of a group $G$ are self normalizing and $G$ satisfies the implication $$ \ o(x)\neq o(y)\Rightarrow o(xy)\neq o(x), o(y), $$ for all non-trivial elements $x$ and $y$, then $G$ is a $p$-group or simple.

自由群循环分裂中的椭圆子群

Elliptic subgroups in cyclic splittings of a free group

本文利用 Whitehead 算法的 Gersten 推广研究自由群初等循环分裂中有限生成子群的椭圆性判定，并给出秩二情形及 HNN-扩张情形的算法结果。

On the finiteness of the classifying space for the family of virtually cyclic subgroups

Given a group G, we consider its classifying space for the family of virtually cyclic subgroups. We show for many groups, including for example, one-relator groups, acylindrically hyperbolic groups, 3-manifold groups and CAT(0) cube groups, that they do not admit a finite model for this classifying space unless they are virtually cyclic. This settles a conjecture due to Juan-Pineda and Leary for these classes of groups.

关于无菱形子群格

On Diamond-Free Subgroup Lattices

该研究定义“循环菱形”子格，刻画有限非循环群必含此结构，并给出无限 Abel 群不含该结构的充要条件。

Conjugacy classes of maximal cyclic subgroups of metacyclic $p$-groups

In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of a finite group $G$. We compute $η(G)$ for all metacyclic $p$-groups. We show that if $G$ is a metacyclic $p$-group of order $p^n$ that is not dihedral, generalized quaternion, or semi-dihedral, then $η(G) \ge n-2$, and we determine when equality holds.

Example of non-linearizable quasi-cyclic subgroup of automorphism group of polynomial algebra

It is well known that every finite subgroup of automorphism group of polynomial algebra of rank 2 over the field of zero characteristic is conjugated with a subgroup of linear automorphisms. We prove that it is not true for an arbitrary torsion subgroup. We construct an example of abelian $p$-group of automorphism of polynomial algebra of rank 2 over the field of complex numbers, which is not conjugated with a subgroup of linear automorphisms.

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.

Counting conjugacy classes of cyclic subgroups for fusion systems

We give another proof of an observation of Thévenaz \cite{T1989} and present a fusion system version of it. Namely, for a saturated fusion system $\CF$ on a finite $p$-group $S$, we show that the number of the $\CF$-conjugacy classes of cyclic subgroups of $S$ is equal to the rank of certain square matrices of numbers of orbits, coming from characteristic bisets, the characteristic idempotent and finite groups realizing the fusion system $\CF$ as in our previous work \cite{P2010}.

Conjugacy classes of maximal cyclic subgroups

In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We consider $η$ and direct and semi-direct products. We characterize the normal subgroups $N$ so that $η(G/N) = η(G)$. We set $G^- = \{ g \in G \mid \langle g \rangle {\rm ~is~not ~maximal~cyclic} \}$. We show if $\langle G^- \rangle < G$, then $G/\langle G^- \rangle$ is either (1) an elementary abelian $p$-group for some prime $p$, (2) a Frobenius group whose Frobenius kernel is a $p$-group of exponent $p$ and a Frobenius complement has order $q$ for distinct primes $p$ and $q$, or (3) isomo…

A note on the number of cyclic subgroups of a finite group

Let $G$ be a finite group, $L_1(G)$ be its poset of cyclic subgroups and consider the quantity $α(G)=\frac{|L_1(G)|}{|G|}$. The aim of this paper is to study the class $\cal{C}$ of finite nilpotent groups having $α(G)=\frac{3}{4}$. We show that if $G$ belongs to this class, then it is a 2-group satisfying certain conditions. Also, we study the appartenance of some classes of finite groups to $\cal{C}$.

具有循环或阿贝尔子群的CLT群

CLT-groups with cyclic or abelian subgroups

本研究对CCLT数与ACLT数进行分类，探究循环/阿贝尔CLT群的性质，证明两类群均属于超可解群类，并在非循环有限群集上引入CCLT度函数研究其性质。

Subgroup Distortion in Wreath Products of Cyclic Groups

We study the effects of subgroup distortion in the wreath products A wr Z, where A is finitely generated abelian. We show that every finitely generated subgroup of A wr Z has distortion function equivalent to some polynomial. Moreover, for A infinite, and for any polynomial l^k, there is a 2-generated subgroup of A wr Z having distortion function equivalent to the given polynomial. Also a formula for the length of elements in arbitrary wreath product H wr G easily shows that the group Z_2 wr Z^2 has distorted subgroups, while the lamplighter group Z_2 wr Z has no distorted (finitely generated)…

GL$(2, q)$的完全可约循环子群的共轭类

Conjugacy classes of completely reducible cyclic subgroups of GL$(2, q)$

本文针对素数幂$q$及不被素数$p$整除的正整数$m$，确定了一般线性群GL$(2,q)$中阶为$m$的完全可约循环子群的共轭类数目。

Finite groups with $\mathbb{P}$-subnormal and strongly permutable subgroups

Let $H$ be a subgroup of a group $G$. The permutizer $P_G(H)$ is the subgroup generated by all cyclic subgroups of $G$ which permute with $H$. A subgroup $H$ of a group $G$ is strongly permutable in $G$ if $P_U(H)=U$ for every subgroup $U$ of $G$ such that~$H\le U\le G$. We investigate groups with $\mathbb{P}$-subnormal or strongly permutable Sylow and primary cyclic subgroups. In particular, we prove that groups with all strongly permutable primary cyclic subgroups are supersoluble.

Computing the Table of Marks of a Cyclic Extension

The subgroup pattern of a finite groups $G$ is the table of marks of $G$ together with a list of representatives of the conjugacy classes of subgroups of $G$. In this article we present an algorithm for the computation of the subgroup pattern of a cyclic extension of $G$ from the subgroup pattern of $G$. Repeated application of this algorithm yields an algorithm for the computation of the table of marks of a solvable group $G$, along a composition series of $G$.

Freiheitssatz for amalgamated products of free groups over maximal cyclic subgroups

In 1930, Wilhelm Magnus introduced the so-called Freiheitssatz: Let $F$ be a free group with basis $\mathcal{X}$ and let $r$ be a cyclically reduced element of $F$ which contains a basis element $x \in \mathcal{X}$, then every non-trivial element of the normal closure of $r$ in $F$ contains the basis element $x$. Equivalently, the subgroup freely generated by $\mathcal{X} \backslash \{x\}$ embeds canonically into the quotient group $F / \langle \! \langle r \rangle \! \rangle_{F}$. In this article, we want to introduce a Freiheitssatz for amalgamated products $G=A \ast_{U} B$ of free groups $A…

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

We study a class $\mathfrak{M}$ of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations. This class includes many finite metacyclic generalized Fibonacci groups that have been previously identified in the literature. By analysing their shift extensions we show that the groups in the class $\mathfrak{M}$ are are coherent, subgroup separable, satisfy the Tits alternative, possess finite index subgroups of geometric dimension at most two, and that their finite subgroups are all metacyclic. Many of t…

The second minimum/maximum value of the number of cyclic subgroups of finite $p$-groups

Let $C(G)$ be the poset of cyclic subgroups of a finite group $G$ and let $\mathcal{P}$ be the class of $p$-groups of order $p^n$ ($n\geq 3$). Consider the function $α:\mathcal{P}\longrightarrow (0, 1]$ given by $α(G)=\frac{|C(G)|}{|G|}$. In this paper, we determine the second minimum value of $α$, as well as the corresponding minimum points. Further, since the problem of finding the second maximum value of $α$ was completely solved for $p=2$, we focus on the case of odd primes and we outline a result in this regard.

有限 $p$-指数子群在多重 HNN 扩张无限循环群中的交集

The intersection of the subgroups of finite $p$-index in a multiple HNN-extension of an infinite cyclic group

本文研究了无限循环群的多重 HNN 扩张中有限 $p$-指数正规子群的交集，并推广了 Moldavanskii 对 Baumslag-Solitar 群的结果。

群的循环子群图

Cyclic Subgroup Graph of a Group

本文研究群的循环子群图的图论性质，分类了使该图为连通、完全、星、树或路径的群，并探讨其与幂图、增强幂图及偏序集的关系。

有限p-群中非循环子群的数量

On the number of non-cyclic subgroups of finite p-groups

本文研究了有限p-群中非循环子群的数量上限，并证明了非交换p-群的该数量上限与特定结构的p-群相关联。

素数幂阶循环群全形的相互正规化正则子群

The mutually normalizing regular subgroups of the holomorph of a cyclic group of prime power order

本文研究有限循环 p-群 $C_{p^n}$ 全形中的相互正规化正则子群，利用 gamma 函数理论构建局部正规化图，以紧凑形式完整刻画这些子群及其关系。

Residual finiteness and strict distortion of cyclic subgroups of solvable groups

We provide polynomial lower bounds for residual finiteness of residually finite, finitely generated solvable groups that admit infinite order elements in the Fitting subgroup of strict distortion at least exponential. For this class of solvable groups which include polycyclic groups with a nontrivial exponential radical and the metabelian Baumslag-Solitar groups, we improve the lower bounds found in the literature. Additionally, for the class of residually finite, finitely generated solvable groups of infinite Prüfer rank that satisfy the conditions of our theorem, we provide the first nontriv…