Finite Metabelian Groups Research Articles

Constrained inhomogeneous spherical equations: average-case hardness

In this paper we analyze computational properties of the Diophantine problem (and its search variant) for spherical equations $\prod_{i=1}^m z_i^{-1} c_i z_i = 1$ (and its variants) over the class of finite metabelian groups $G_{p,n}=\mathbb{Z}_p^n \rtimes \mathbb{Z}_p^\ast$, where $n\in\mathbb{N}$ and $p$ is prime. We prove that the problem of finding solutions for certain constrained spherical equations is computationally hard on average (assuming that some lattice approximation problem is hard in the worst case).

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.

On Some Graphs of Finite Metabelian Groups of Order Less Than 24

In this work, a non-abelian metabelian group is represented by G while represents conjugacy class graph. Conjugacy class graph of a group is that graph associated with the conjugacy classes of the group. Its vertices are the non-central conjugacy classes of the group, and two distinct vertices are joined by an edge if their cardinalities are not coprime. A group is referred to as metabelian if there exits an abelian normal subgroup in which the factor group is also abelian. It has been proven earlier that 25 non-abelian metabelian groups which have order less than 24, which are considered in this work, exist. In this article, the conjugacy class graphs of non-abelian metabelian groups of order less than 24 are determined as well as examples of some finite groups associated to other graphs are given.

Finite Metabelian Group Algebras

Given a finite metabelian group G, whose central quotient is abelian (not cyclic) group of order p2, p odd prime, the objective of this paper is to obtain a complete algebraic structure of semisimple group algebra Fq[G] in terms of primitive central idempotents, Wedderburn decomposition and the automorphism group.

The Algebraic Structure of Finite Metabelian Group Algebras

An algorithm for the explicit computation of a complete set of primitive central idempotents, Wedderburn decomposition, and the automorphism group of the semisimple group algebra of a finite metabelian group is developed. The algorithm is illustrated with its application to the semisimple group algebra of an arbitrary metacyclic group, and certain indecomposable groups whose central quotient is the Klein four-group.

Primitive and measure-preserving systems of elements on the varieties of metabelian and metabelian profinite groups

Given a free metabelian group S of finite rank r, r ≥ 2, we prove that a system of elements g1, ..., gn ∈ S for n = 1 or n = r preserves measure on the variety of all metabelian groups if and only if the system is primitive. Similar results hold for a free profinite group \(\hat S\) and the variety of finite metabelian groups for each n, 1 ≤ n ≤ r. Some corollaries to these theorems are derived.

THE RATIONAL GROUP ALGEBRA OF A FINITE GROUP

An explicit expression for the primitive central idempotent of the rational group algebra ℚ[G] of a finite group G associated with any complex irreducible character of G is obtained. A complete set of primitive central idempotents and the Wedderburn decomposition of the rational group algebra of a finite metabelian group is also computed.

Problems of classifying associative or Lie algebras over a field of characteristic not two and finite metabelian groups are wild

Let F be a field of characteristic different from 2. It is shown that the problems ofclassifying    (i) local commutative associative algebras over F with zero cube radical,    (ii) Lie algebras over F with central commutator subalgebra of dimension 3, and    (iii) finite p-groups of exponent p with central commutator subgroup of order p3are hopeless since each of them contains     • the problem of classifying symmetric bilinear mappings U × U → V , or     • the problem of classifying skew-symmetric bilinear mappings U × U → V ,in which U and V are vector spaces over F (consisting of p elements for p-groups (iii)) and V is 3-dimensional. The latter two problems are hopeless since they are wild; i.e., each of them contains the problem of classifying pairs of matrices over F up to similarity.

A construction of metabelian groups

In 1934, Garrett Birkhoff has shown that the number of isomorphism classes of finite metabelian groups of order $p^{22}$ tends to infinity with $p$. More precisely, for each prime number $p$ there is a family $(M_\lambda)_{\lambda=0,...,p-1}$ of indecomposable and pairwise nonisomorphic metabelian $p$-groups of the given order. In this manuscript we use recent results on the classification of possible embeddings of a subgroup in a finite abelian $p$-group to construct families of indecomposable metabelian groups, indexed by several parameters, which have upper bounds on the exponents of the center and the commutator subgroup.

The normalizer of a metabelian group in its integral group ring

In this paper, we discuss two cases where all C -automorphisms are inner; therefore, the normalizer property holds for those cases. Our results generalize a result of Marciniak and Roggenkamp. As an application of our theorems, we prove that the normalizer property holds for the integral group ring of a split finite metabelian group with a dihedral Sylow 2-subgroup. Our proofs rely heavily on the method of Marciniak and Roggenkamp.

A Note on Finite Metabelian Groups of Wielandt Length Two

The class of groups of Wielandt length one has been extensively studied and recently the class of groups of Wielandt length two has been studied, in particular nilpotent groups of odd order by E.A. Ormerod and supersoluble groups of order prime to six by A. Ali. In this paper we consider metabelian groups of odd order and provide characterizations for those groups with nilpotent residual either a Hall subgroup or of prime power order.

A Note on Finite Metabelian Groups of Wielandt Length Two

A Note on Finite Metabelian Groups of Wielandt Length Two

A FURTHER CLASS OF 3-GENERATOR, 3-RELATION FINITE GROUPS

We exhibit a further class of 3-generator, 3-relation finite metabelian groups.

A Constructive Brauer-Witt Theorem for Certain Solvable Groups

AbstractDivision algebras occurring in simple components of group algebras of finite groups over algebraic number fields are studied. First, well-known restrictions are presented for the structure of a group that arises once no further Clifford Theory reductions are possible. For groups with these properties, a character-theoretic condition is given that forces the p-part of the division algebra part of this simple component to be generated by a predetermined p-quasi-elementary subgroup of the group, for any prime integer p. This is effectively a constructive Brauer-Witt Theorem for groups satisfying this condition. It is then shown that it is possible to constructively compute the Schur index of a simple component of the group algebra of a finite nilpotent-by-abelian group using the above reduction and an algorithm for computing Schur indices of simple algebras generated by finite metabelian groups.

The Structure of Simple Rings Generated by Finite Metabelian Groups

The Structure of Simple Rings Generated by Finite Metabelian Groups

On the field of a 2-block. II

Every real 2 2 -block B B of a finite metabelian group contains an irreducible character θ \theta such that Q ( B ) = Q ( θ ) Q(B) = Q(\theta ) .

Rational Nonlinear Characters of Metabelian Groups

Let $G$ be a finite metabelian group with all nonlinear irreducible characters rational. Then the exponent of the commutator group $G’$ is a prime or divides 16, 24, or 40. If $G’$ is also cyclic, then its order is a prime or divides 12.

Rational nonlinear characters of metabelian groups

Let G G be a finite metabelian group with all nonlinear irreducible characters rational. Then the exponent of the commutator group G ′ G’ is a prime or divides 16, 24, or 40. If G ′ G’ is also cyclic, then its order is a prime or divides 12.

On characters of height zero

AbstractEvery irreducible ordinary character in a p-block of a finite metabelian group is of height 0 if and only if the defect group of the p-block is abelian.

Finite potent groups

A group is potent if for any element of the group and any prescribed positive integer (dividing its order if this order is finite) there corresponds a finite homomorphic image of the group in which the element has the prescribed integer as its order. The finite potent groups form a finite variety that contains all finite nilpotent groups, all finite metabelian groups, and precisely one simple group, A5.