Split Metacyclic Groups Research Articles

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.

Cohomology ring of the finite split metacyclic group

Let G be the finite split metacyclic group of presentation G=<A, B | Am=Bn=e, BAB-1= Ar, rn ≡ 1(m), (n(r-1), m)=1>. Huebschmann computed the integral cohomology ring of a large class of metacyclic groups using the mechanism of homological perturbation theory and determined the generators of the subring of even degree classes of any metacyclic group by means of the Chern classes of some of its representations. The purpose of this article is to investigate H∗(G; ℤ) by localization methods. It also aims to determine explicit representations of G whose Chern classes generate H∗(G; ℤ) by induction techniques.

Good codes from metacyclic groups II

In this paper, we consider semisimple group algebras [Formula: see text] of split metacyclic groups over finite fields. We construct left codes in [Formula: see text] in the case when the order [Formula: see text] is [Formula: see text], where [Formula: see text] and [Formula: see text] are different primes such that [Formula: see text] extend the construction described in a previous paper, determine their dual codes and find some good codes.

Metric dimension of Cayley digraphs of split metacyclic groups

A directed Cayley graph Cay(Γ,X) is specified by a group Γ and an identity-free generating set X for this group. Vertices of Cay(Γ,X) are elements of Γ and there is a directed edge from a vertex u to a vertex v in Cay(Γ,X) if and only if there is a generator x∈X such that ux=v. We study the metric dimension for the directed Cayley graphs Cay(Γs,{a,b}) of general split metacyclic groups, and present the exact values of the metric dimension for the special split metacyclic groups Γs=〈a,b|an=b2s=1,ba=a−1b〉.

Bound on the diameter of metacyclic groups

Let [Formula: see text] be the split metacyclic group, where [Formula: see text] is a unit modulo [Formula: see text]. We derive an upper bound for the diameter of [Formula: see text] using an arithmetic parameter called the weight, which depends on [Formula: see text], [Formula: see text], and the order of [Formula: see text]. As an application, we show how this would determine a bound on the diameter of an arbitrary metacyclic group.

Absolutely split metacyclic groups and weak metacirculants

Let [Formula: see text] be positive integers, and let [Formula: see text] be a split metacyclic group such that [Formula: see text]. We say that [Formula: see text] is absolutely split with respect to[Formula: see text] provided that for any [Formula: see text], if [Formula: see text], then there exists [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we give a sufficient and necessary condition for the group [Formula: see text] being absolutely split. This generalizes a result of Sanming Zhou and the second author in [Weak metacirculants of odd prime power order, J. Comb. Theory A 155 (2018) 225–243]. We also use this result to investigate the relationship between metacirculants and weak metacirculants. Metacirculants were introduced by Alspach and Parsons in [Formula: see text] and have been a rich source of various topics since then. As a generalization of this class of graphs, Marušič and Šparl in 2008 introduced the so-called weak metacirculants. A graph is called a weak metacirculant if it has a vertex-transitive metacyclic automorphism group. In this paper, it is proved that a weak metacirculant of [Formula: see text]-power order is a metacirculant if and only if it has a vertex-transitive split metacyclic automorphism group. This provides a partial answer to an open question in the literature.

A note on normal complement problem for split metacyclic groups

In this article, we discuss the normal complement problem for metacyclic groups in modular group algebras. If F is the field with p elements and G is a finite split metacyclic p-group of nilpotency class 2, then we prove that G has a normal complement in For a finite field F of characteristic p, where p is an odd prime, we prove that has a normal complement in if and only if p = 3 and

Commuting Graphs of Split Metacyclic Groups

We study the link between groups and graphs created by considering the commuting graph of a group. We focus our efforts on groups that can be represented as the semidirect product of cyclic groups, and we describe the commuting graphs of two classes of such groups.

On the irreducibility of the space of Galois covers of curves of a given numerical type for certain split metacyclic groups

We consider the spaces Mg(G)⊂Mg of curves of genus g≥2, that admit an effective action by a given finite group G and want to understand its irreducible components. We prove that if G=Cm⋊Cn is a split metacyclic group where both factors have odd prime order, the irreducible components are determined by a numerical datum which comes from the monodromy of the induced G-coverings.

Generators of split extensions of Abelian groups by cyclic groups

Let G \simeq M \rtimes C be an n -generator group which is a split extension of an Abelian group M by a cyclic group C . We study the Nielsen equivalence classes and T-systems of generating n -tuples of G . The subgroup M can be turned into a finitely generated faithful module over a suitable quotient R of the integral group ring of C . When C is infinite, we show that the Nielsen equivalence classes of the generating n -tuples of G correspond bijectively to the orbits of unimodular rows in M^{n -1} under the action of a subgroup of GL _{n - 1}(R) . Making no assumption on the cardinality of C , we exhibit a complete invariant of Nielsen equivalence in the case M \simeq R . As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag–Solitar groups, split metacyclic groups and lamplighter groups.

Group algebras of metacyclic groups over finite fields

In this paper, we consider semisimple group algebras \({\mathbb {F}}_{q}G\) of non abelian split metacyclic groups over a finite field. We give conditions for them to have a minimal number of simple components and find the primitive central idempotents of \({\mathbb {F}}_{q}G\) in the case when the order G is equals \(p^{m}\ell ^{n}\), where p and \(\ell \) are different prime numbers.

CHARACTER TABLES OF METACYCLIC GROUPS

AbstractWe show that two metacyclic groups of the following types are isomorphic if they have the same character tables: (i) split metacyclic groups, (ii) the metacyclic p-groups and (iii) the metacyclic {p, q}-groups, where p, q are odd primes.

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.

Cellular decomposition and free resolution for split metacyclic spherical space forms

Given a free isometric action of a split metacyclic group on odd dimensional sphere, we obtain an explicit finite cellular decomposition of the sphere equivariant with respect to the group action. A cell decomposition of the factor space and an explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group follow. In particular, the construction provides a simple explicit 4-periodic free resolution for the split metacyclic groups.

On automorphisms of split metacyclic groups

Let D(m, n; k) be the semi-direct product of two finite cyclic groups \({\mathbb{Z}/m=\langle x\rangle}\) and \({\mathbb{Z}/n=\langle y\rangle}\) , where the action is given by yxy −1 = x k . In particular, this includes the dihedral groups D 2m . We calculate the automorphism group Aut (D(m, n; k)).

Genus spectra for split metacyclic groups

An integer n\geq2 is said to be a genus of a finite group G if there is a compact Riemann surface of genus n on which G acts as a group of automorphisms. In this paper, formulae are given for the minimum genus, minimum stable genus and the gap sequence, i.e., the (finite) set of non-genera, for a split metacyclic group of order pq, where p and q are primes. This information completely determines the genus spectrum for such groups.

The nonabelian tensor square of a finite split metacyclic group

Given any group G, its tensor square G ⊗ G is defined by the followingpresentation(see [[3]):where g,g′,h,h′ range independently over G, and gh=ghg−1. In what follows, gg′ ⊗ gh is often written in the abbreviated form g(g′ ⊗ h).

Locally isomorphic algebras and a Hasse principle for split metacyclic groups

Locally isomorphic algebras and a Hasse principle for split metacyclic groups

Real representations of split metacyclic groups

Real representations of split metacyclic groups