The Energy of Conjugate Graph of Some Finite Metabelian Groups of Order Less Than 24

The conjugacy classes of the Metabelian group G, plays an important role in defining the conjugate graph, whose vertices are non-central elements of G, and two vertices are connected if and only if they are conjugate. The constructions of conjugate graphs of all non abelian metabelian groups of order less than 24 are the basis for this paper. And the obtained results are then used to calculate the energy of the aforementioned group. This is aided by specialized programming software (maple).

Dehn functions of finitely presented metabelian groups

In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.

Groups with finitely many non-isomorphic factor-groups

We prove that if [Formula: see text] is either a hypercentral-by-finite group or a soluble Baer group and if [Formula: see text] has finitely many non-isomorphic factor-groups, then [Formula: see text] is a Chernikov group. The converse is also true. Furthermore, we give some information on the structure of a metabelian group with finitely many non-isomorphic factor-groups.

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.

Parallel identity testing for skew circuits with big powers and applications

Powerful skew arithmetic circuits are introduced. These are skew arithmetic circuits with variables, where input gates can be labeled with powers [Formula: see text] for binary encoded numbers [Formula: see text]. It is shown that polynomial identity testing for powerful skew arithmetic circuits belongs to [Formula: see text], which generalizes a corresponding result for (standard) skew circuits. Two applications of this result are presented: (i) Equivalence of higher-dimensional straight-line programs can be tested in [Formula: see text]; this result is even new in the one-dimensional case, where the straight-line programs produce words. (ii) The compressed word problem (or circuit evaluation problem) for certain wreath products of finitely generated abelian groups belongs to [Formula: see text]. Using the Magnus embedding, it follows that the compressed word problem for a free metabelian group belongs to [Formula: see text].