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).

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.

Unit group of semisimple group algebras of some non-metabelian groups of order 120

In this paper, we give the characterization of the unit groups of semisimple group algebras of some non-metabelian groups of order 120. This study completes the study of unit groups of semisimple group algebras of all groups up to order 120, except that of the symmetric group [Formula: see text] and groups of order 96.

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.

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.