Finite Metacyclic Groups Research Articles

The isomorphism problem for rational group algebras of finite metacyclic nilpotent groups

. We prove that if G and H are finite metacyclic groups with isomorphic rational group algebras and one of them is nilpotent, then G and H are isomorphic.

Triple exterior products and 2-nilpotent multipliers of finite split metacyclic groups

There has been a great importance in understanding the nilpotent multipliers of finite groups in recent past. Let a group G be presented as the quotient of a free group F by a normal subgroup R. Given a positive integer c, the c-nilpotent multiplier of the group G is the abelian group M ( c ) ( G ) = ( R ∩ γ c + 1 ( F ) ) / γ c + 1 ( R , F ) . In particular, M ( 1 ) ( G ) is the Schur multiplier of G. The study of Schur multiplier of finite metacyclic groups goes back to the paper by F. R. Beyl in 1973. In this article, we study the 2-nilpotent multiplier of finite split metacyclic groups.

Degrees of faithful irreducible representations of metabelian groups

In 1993, Sim proved that all the faithful irreducible representations of a finite metacyclic group over any field of positive characteristic have the same degree. In this paper, we restrict our attention to non-modular representations and generalize this result for — (1) finite metabelian groups, over fields of positive characteristic coprime to the order of groups, and (2) finite groups having a cyclic quotient by an abelian normal subgroup, over number fields.

Commutation semigroups of finite metacyclic groups with trivial centre

We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented $$\begin{aligned} G(m,n,k) = \left\langle {a,b;{a^m} = 1,{b^n} = 1,{a^b} = {a^k}} \right\rangle \quad (m,n,k\in {\mathbb {Z}}^+) \end{aligned}$$with $$(m,k - 1) = 1$$ and $$n = \mathrm {ind}_m(k),$$ the smallest positive integer for which $${k^n} = 1\,\pmod m,$$ with the conjugate of a by b written $${a^b} = {b^{ - 1}}ab.$$ The right and left commutation semigroups ofG, denoted $$\mathrm{P}(G)$$ and $$\Lambda (G),$$ are the semigroups of mappings generated by $$\rho (g):G \rightarrow G$$ and $$\lambda (g):G \rightarrow G$$ defined by $$(x)\rho (g) = [x,g]$$ and $$(x)\lambda (g) = [g,x],$$ where the commutator of g and h is defined as $$[g,h] = {g^{ - 1}}{h^{ - 1}}gh.$$ This paper builds on a previous study of commutation semigroups of dihedral groups conducted by the authors with C. Levy. Here we show that a similar approach can be applied to G, a metacyclic group with trivial centre. We give a construction of $$\mathrm{P}(G)$$ and $$\Lambda (G)$$ as unions of containers, an idea presented in the previous paper on dihedral groups. In the case that $$\left\langle a \right\rangle$$ is cyclic of order p or $${p^2}$$ or its index is prime, we show that both $$\mathrm{P}(G)$$ and $$\Lambda (G)$$ are disjoint unions of maximal containers. In these cases, we give an explicit representation of the elements of each commutation semigroup as well as formulas for their exact orders. Finally, we extend a result of J. Countryman to show that, for G(m, n, k) with m prime, the condition $$\left| {\mathrm{P}(G)} \right| = \left| {\Lambda (G)} \right|$$ is equivalent to $$\mathrm{P}(G) = \Lambda (G).$$

The Subgroups of Finite Metacyclic Groups

In this paper, the author characterizes the subgroups of a finite metacyclic group K by building a one to one correspondence between certain 3-tuples (k,l,β) ∈ ℕ3 and all the subgroups of K. The results are applied to compute some subgroups of K as well as to study the structure and the number of p-subgroups of K, where p is a fixed prime number. In addition, the author gets a factorization of K, and then studies the metacyclic p-groups, gives a different classification, and describes the characteristic subgroups of a given metacyclic p-group when p ≥ 3. A “reciprocity” relation on enumeration of subgroups of a metacyclic group is also given.

Diagonal resolutions for metacyclic groups

We show the finite metacyclic groups G(p,q) admit a class of projective resolutions which are periodic of period 2q and which in addition possess the properties that a) the differentials are 2×2diagonal matrices; b) the Swan–Wall finiteness obstruction (cf. [19,20]) vanishes. We obtain thereby a purely algebraic proof of Petrie's Theorem ([14]) that G(p,q) has free period 2q.

Finite metacyclic groups as active sums of cyclic subgroups

The notion of active sum provides an analogue for groups of what the direct sum is for abelian groups. One natural question then is which groups are the active sum of a family 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.

Computing Irreducible Representation of Finite Metacyclic Groups of Order 16 Using Burnside Method

Irreducible representation is the nucleus of a character table and is of great importance in chemistry. This paper focuses on finite metacyclic groups and their irreducible representation. This study aims to find out the irreducible representation of finite metacyclic groups of class two and finite metacyclic group of class at least three of negative type that can have order 16 by using Burnside method. Keywords : Irreducible representation; finite metacyclic groups; Burnside method. 2010 Mathematics Subject Classification 20G05; 20C99

An Example on Computing the Irreducible Representation of Finite Metacyclic Groups by Using Great Orthogonality Theorem Method

Representation theory is a study of real realizations of the axiomatic systems of abstract algebra. For any group, the number of possible representative sets of matrices is infinite, but they can all be reduced to a single fundamental set, called the irreducible representations of the group. This paper focuses on an example of finite metacyclic groups of class two of order 16. The irreducible representation of that group is found by using Great Orthogonality Theorem Method.

On split metacyclic groups

We consider sufficient conditions for metacyclic groups to split. Specifically, we show that a finite metacyclic group G of odd order is split on its cyclic normal subgroup K if K is such that G/K is cyclic and |K| = exp G.

Ring Isomorphism of Cyclic Cyclotomic Algebras

It is shown that ring isomorphisms between cyclic cyclotomic algebras over cyclotomic number fields are essentially determined by the list of local Schur indices at all rational primes. As a consequence, ring isomorphisms between simple components of the rational group algebras of finite metacyclic groups are determined by the center, the dimension over ℚ, and the list of local Schur indices at rational primes. An example is given to show that this does not hold for finite groups in general.

Central Units in Metacyclic Integral Group Rings

In this article, we give a method to compute the rank of the subgroup of central units of ℤ G, for a finite metacyclic group, G, by means of ℚ-classes and ℝ-classes. Then we construct a multiplicatively independent set 𝒰 ⊂ 𝒵(U(ℤ C p, q )) and by applying our results, we prove that 𝒰 generates a subgroup of finite index.

Mini-Workshop: Arithmetik von Gruppenringen

The mini workshop "Arithmetic of group rings" was attended by 16 participants from Belgium, Brazil, Canada, Germany, Hungary, Israel, Italy, Romania and Spain. The expertise was a good mixture between senior and young researchers. It was a very stimulating experience and thesize of the group allowed excellent discussions amongst all participants. Very fruitful were the problem sessions, resulting in the problems listed at the end of this report. The main highlights of the conference were: The complete calculation of the projective Schur subgroup of the Brauer group by Aljadeff and del Rio. Hertweck's solution of the first Zassenhaus conjecture for finite metacyclic groups. the description of special subgroups of the unit group of integral group rings, such as the hypercentre and the finite conjugacy centre, and the relation with respect to the normalizer of the trivial units. discussion of the present state of art via several survey talks presented and the problem sessions The group G determines its integral group ring \mathbb Z G and its group V(\mathbb Z G) of normalized units. Several talks addressed the interplay of the cohomological properties of these three objects. Further topics included twisted group rings, group rings over local rings, polynomial growth and identities, orders and semigroup rings, Lie structure, representation-theoretic and algorithmic methods.

The Group of Automorphisms of the Rational Group Algebra of a Finite Metacyclic Group

We investigate the group of automorphisms Aut(ℚ G) of a rational group algebra ℚ G of a finite metacyclic group G by first describing the simple components of the Wedderburn decomposition of ℚ G and then investigating when two of these simple components are isomorphic.

Twisted group rings of metacyclic groups

Given a finite metacyclic group G G , a central extension F F having the projective lifting property over all fields is constructed. This extension and its group rings are used to investigate the faithful irreducible projective representations of G G and the fields over which they can be realized. A full description of the finite metacyclic groups having central simple twisted group rings over fields is given.

FINITE METACYCLIC GROUPS WITH FAITHFUL IRREDUCIBLE REPRESENTATIONS

A metacyclic group is a group that has a cyclic normal subgroup with cyclic quotient. We characterize finite metacyclic groups that have faithful irreducible representations over a given field.

Finite metacyclic groups acting on bordered surfaces

A group is called metacyclic in case both its commutator subgroup and commutator quotient group are cyclic. Thus a metacyclic group is a cyclic extension of a cyclic group, and metacyclic groups are among the best understood of the nonabelian groups. Many interesting groups are metacyclic. For instance, the dihedral groups and the “odd” dicyclic groups are metacyclic; see [4, pp. 9–11] for more examples. Here we shall consider the actions of these groups on bordered Klein surfaces.

The non-orientable genus of some metacyclic groups

We describe non-orientable, octagonal embeddings for certain 4-valent, bipartite Cayley graphs of finite metacyclic groups, and give a class of examples for which this embedding realizes the non-orientable genus of the group. This yields a construction of Cayley graphs for which\(2\gamma - \tilde \gamma \) is arbitrarily large, where γ and\(\tilde \gamma \) are the orientable genus and the non-orientable genus of the Cayley graph.

A Comment on Finite Nilpotent Groups of Deficiency Zero

A finite group is said to have deficiency zero if it can be presented with an equal number of generators and relations. Finite metacyclic groups of deficiency zero have been classified, see [1] or [6]. Finite non-metacyclic groups of deficiency zero, which we denote by FD0-groups, are relatively scarce. In [3] I. D. Macdonald introduced a class of nilpotent FD0-groups all having nilpotent class≤8. The largest nilpotent class known for a Macdonald group is 7 [4]. Only a finite number of nilpotent FD0-groups, other than the Macdonald groups, seem to be known [5], [7]. In this note we exhibit a class of FD0-groups which contains nilpotent groups of arbitrarily large nilpotent class.

Finite metacyclic groups all extensions of which split

SynopsisThe groups of the title are characterized arithmetically.