A group G is almost cyclic if there is an element x in G, such that for all g in G, there is an element y in G and an integer n with ygy^{-1} = x^n (that is, every element is conjugate to some power of x). W. Ziller asked whether there are finitely-presented almost cyclic groups which are not cyclic in connection with work on closed geodesics. V. Guba constructed an infinite (non-cyclic) finitely generated almost cyclic group. The principal results of this paper are: Solvable almost cyclic groups are cyclic, and one-relator almost cyclic groups are cyclic.
This paper is a significant part of a general project aimed to classify all irreducible representations of finite quasi-simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix. (A square matrix $M$ is called almost cyclic if it is similar to a block-diagonal matrix with two blocks, such that one block is scalar and another block is a matrix whose minimum and characteristic polynomials coincide. Reflections and transvections are examples of almost cyclic matrices. The paper focuses on the Weil representations of fini…
This paper is a significant contribution to a general programme aimed to classify all projective irreducible representations of finite simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix (that is, a square matrix $M$ of size $n$ over a field $F$ with the property that there exists $α\in F$ such that $M$ is similar to $diag(α\cdot Id_k, M_1)$, where $M_1$ is cyclic and $0\leq k\leq n$). While a previous paper dealt with the Weil representations of finite classical groups, which play a key role in the general pict…
We consider groups defined by cyclic presentations where the defining word has length three and the cyclic presentation satisfies the T(6) small cancellation condition. We classify when these groups are hyperbolic. When combined with known results, this completely classifies the hyperbolic T(6) cyclically presented groups.
The cyclic graph of a group $G$ is the graph whose vertices are the nonidentity elements of $G$ and whose edges connect distinct elements $x$ and $y$ if and only if the subgroup $\langle x,y\rangle$ is cyclic. We obtain information about the cyclic graph of $2$-Frobenius groups. The cyclic graph of a $2$-Frobenius group is disconnected. In this paper, we determine the number of connected components of the cyclic graph of any $2$-Frobenius group.
All crossed products of two cyclic groups are explicitly described using generators and relations. A necessary and sufficient condition for an extension of a group by a group to be a cyclic group is given.
Garside groups are a natural lattice-theoretic generalisation of the braid groups and spherical type Artin--Tits groups. Here we show that the class of Garside groups is closed under some free products with cyclic amalgamated subgroups. We deduce that every tree product of infinite cyclic groups is a Garside group. Moreover, we study those cyclic HNN extensions of Garside groups that are Garside groups as well. Using a theorem of Pietrowski, we conclude this paper by stating that a non-cyclic one-relator group is Garside if and only if its centre is nontrivial.
Marius Tărnăuceanu described the finite groups $G$ having $|G|-1$ cyclic subgroups. We describe the finite groups $G$ having $|G|-Δ$ cyclic subgroups for $Δ=2, 3, 4$ and $5$.
In this article we give an order-dividing bijective function between cyclic and non cyclic groups of finite order. In particular, we prove that there exists a bijective function from D_{2n} to Z_{2n} for any natural integer n; and from Z_p x Z_k to Z_{pk} when p is an odd prime and k is not a multiple of p.
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.
In this paper we introduce and study the concept of cyclic factorization number of a finite group G. By using the Mobius inversion formula and other methods involving the cyclic subgroup structure, this is explicitly computed for some important classes 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.
We show that certain graphs of groups with cyclic edge groups are aTmenable. In particular, this holds when each vertex group is either virtually special or acts properly and semisimply on $\mathbb{H}^n$.
Hans Zassenhaus conjectured that every torsion unit of the integral group ring of a finite group $G$ is conjugate within the rational group algebra to an element of the form $\pm g$ with $g\in G$. This conjecture has been disproved recently for metabelian groups, by Eisele and Margolis. However it is known to be true for many classes of solvable groups, as for example nilpotent groups, cyclic-by-abelian groups and groups having a cyclic Sylow subgroup with abelian complement. On the other hand, it is not known whether the conjecture holds for supersolvable groups. This paper is a contribution …
In this article we study the extent to which an $n$-dimensional compact flat manifold with the cyclic holonomy group of square-free order may be distinguished by the finite quotients of its fundamental group. In particular, we display a formula for the cardinality of profinite genus of the fundamental group of an $n$-dimensional compact flat manifold with the cyclic holonomy group of square-free order.
We show that for any $n$-ary group $(G,f)$, the group $Aut(G,f)$ can be embedded in $Aut(\mathbb{Z}_{n-1}\ltimes G)$ and so we can obtain a class of interesting automorphisms of cyclic extensions.
A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We characterise the orientable, non-orientable, and redundant cyclic presentations and obtain concise refinements of these presentations. We show that the Tits alternative holds for the class of groups defined by redundant cyclic presentations and that if the number of generators of the cyclic presentation is greater than two then the corresponding group is large. Generalizing and extending earlier results of the authors we describe the star graphs of ori…
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 $[0, 1]$.
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.
Tărnăuceanu described the finite groups $G$ having exactly $|G|-1$ cyclic subgroups. In "Finite Groups with a Prescribed Number of Cyclic Subgroups,", we used elementary methods to completely characterize those finite groups $G$ having exactly $|G|-Δ$ cyclic subgroups for $Δ=2, 3, 4$ and $5$. In this paper, we prove that for any $Δ>0$ if $G$ has exactly $|G|-Δ$ cyclic subgroups, then $|G|\le 8Δ$ and therefore the number of such $G$ is finite. We then use the computer program GAP to find all $G$ with exactly $|G|-Δ$ cyclic subgroups for $Δ=1,\ldots,32$.
The first author, in 2001, presented a structure for the Baer-invariant of a nilpotent product of cyclic groups with respect to the variety of nilpotent groups. In this paper, using the above structure, we prove the existence of an ${\cal N}_c$-covering group for a nilpotent product of a family of cyclic groups. We also present a structure for the ${\cal N}_c$-covering group.
We consider groups of the form $G=AB$ with two locally cyclic subgroups $A$ and $B$. The structure of these groups is determined in the cases when $A$ and $B$ are both periodic or when one of them is periodic and the other is not. Together with a previous study of the case when $A$ and $B$ are torsion-free, this gives a complete classification of all groups that are the product of two locally cyclic groups. It is also shown that the product of a finite number of pairwise permutable periodic locally cyclic subgroups is a locally supersoluble group.
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.
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.
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.
A group $G$ is called logically cyclic, if it contains an element $s$ such that every element of $G$ can be defined by a first order formula with parameter $s$. The aim of this paper is to investigate the structure of such groups.
Every semidirect product of groups $K\rtimes H$ has size $\left | K \right | \cdot \left | H\right |$, yet the size of such a group's automorphism group varies with the chosen action of $H$ on $K$. This paper will explore groups of the form $\text{Aut}(K\rtimes H)$, considering especially the case where $H$ and $K$ are cyclic. Only finite groups will be considered.
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.
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 $G$.
Let $H$ be a subgroup of a group $G$. We say that $H$ satisfies the power condition with respect to $G$, or $H$ is a power subgroup of $G$, if there exists a non-negative integer $m$ such that $H=G^{m}=<g^{m} | g \in G >$. In this note, the following theorem is proved: Let $G$ be a group and $k$ the number of non-power subgroups of $G$. Then (1) $k=0$ if and only if $G$ is a cyclic group(theorem of F. Sz$\acute{a}$sz) ;(2) $0 < k <\infty$ if and only if $G$ is a finite non-cyclic group; (3) $k=\infty$ if and only if $G$ is a infinte non-cyclic group. Thus we get a new criterion for the finite …
We give a necessary and sufficient condition for the fundamental group of a finite graph of groups with infinite cyclic edge groups to be acylindrically hyperbolic, from which it follows that a finitely generated group splitting over Z cannot be simple. We also give a necessary and sufficient condition (when the vertex groups are torsion free) for the fundamental group to be balanced, where a group is said to be balanced if $x^m$ conjugate to $x^n$ implies that $|m|=|n|$ for all infinite order elements $x$.
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.
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…
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.
We study the conjugacy problem in cyclic extensions of free groups. It is shown that the conjugacy problem is solvable in split extensions of finitely generated free groups by virtually inner automorphisms. An algorithm for construction of the unique representative (the conjugacy normal form) for each conjugacy class is given.
Let $G$ be a finite group having a factorisation $G=AB$ into subgroups $A$ and $B$ with $B$ cyclic and $A\cap B=1,$ and let $b$ be a generator of $B$. The associated skew-morphism is the bijective mapping $f:A \to A$ well defined by the equality $baB=f(a)B$ where $a\in A$. In this paper, we shall classify all skew-morphisms of cyclic $p$-groups where $p$ is an odd prime.
We give a geometric proof of a well known theorem that describes splittings of a free group as an amalgamated product or HNN extension over the integers. The argument generalizes to give a similar description of splittings of a virtually free group over a virtually cyclic 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.
We introduce a condition on the monodromy of a free-by-cyclic group, $G_ϕ$, called the chain flare condition, that implies that the $L^2$-torsion, $ρ^{(2)}(G_ϕ)$, is non-zero. We conjecture that this condition holds whenever the monodromy is exponentially growing.
