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.

Metabelian groups: Full-rank presentations, randomness and Diophantine problems

We study metabelian groups 𝐺 given by full rank finite presentations \langle A\mid R\rangle_{\mathcal{M}} in the variety ℳ of metabelian groups. We prove that 𝐺 is a product of a free metabelian subgroup of rank \max\{0,\lvert A\rvert-\lvert R\rvert\} and a virtually abelian normal subgroup, and that if \lvert R\rvert\leq\lvert A\rvert-2, then the Diophantine problem of 𝐺 is undecidable, while it is decidable if \lvert R\rvert\geq\lvert A\rvert. We further prove that if \lvert R\rvert\leq\lvert A\rvert-1, then, in any direct decomposition of 𝐺, all factors, except one, are virtually abelian. Since finite presentations have full rank asymptotically almost surely, metabelian groups finitely presented in the variety of metabelian groups satisfy all the aforementioned properties asymptotically almost surely.

Electric-magnetic duality in twisted quantum double model of topological orders

We derive a partial electric-magnetic (PEM) duality transformation of the twisted quantum double (TQD) model TQD(G, α) — discrete Dijkgraaf-Witten model — with a finite gauge group G, which has an Abelian normal subgroup N , and a three-cocycle α ∈ H3(G, U(1)). Any equivalence between two TQD models, say, TQD(G, α) and TQD(G′, α′), can be realized as a PEM duality transformation, which exchanges the N-charges and N-fluxes only. Via the PEM duality, we construct an explicit isomorphism between the corresponding TQD algebras Dα(G) and Dα′(G′) and derive the map between the anyons of one model and those of the other.

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.

Critical maximal subgroups and conjugacy of supplements in finite soluble groups

LetGbe a finite group with an abelian normal subgroupN. When doesNhave a unique conjugacy class of complements inG? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups ofGclosed under conjugation whose intersection equals{\Phi(G)}. In particular, we characterize the soluble groups having a unique minimal family with this property (Theorem 2.3, Remark 2.4). In the case when{\Phi(G)=1}, these are exactly the soluble groups in which each abelian normal subgroup has a unique conjugacy class of complements.

GROUP ALGEBRAS WITH ENGEL UNIT GROUPS

Let $F$ be a field of characteristic $p\geq 0$ and $G$ any group. In this article, the Engel property of the group of units of the group algebra $FG$ is investigated. We show that if $G$ is locally finite, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G$ is locally nilpotent and $G^{\prime }$ is a $p$-group. Suppose that the set of nilpotent elements of $FG$ is finite. It is also shown that if $G$ is torsion, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G^{\prime }$ is a finite $p$-group and $FG$ is Lie Engel, if and only if ${\mathcal{U}}(FG)$ is locally nilpotent. If $G$ is nontorsion but $FG$ is semiprime, we show that the Engel property of ${\mathcal{U}}(FG)$ implies that the set of torsion elements of $G$ forms an abelian normal subgroup of $G$.

Sharply 2-transitive groups

We give an explicit construction of infinite sharply 2-transitive groups with fixed point free involutions and without nontrivial abelian normal subgroup.

Noether's Problem for p-Groups with an Abelian Subgroup of Index p

Let K be a field and G be a finite group. Let G act on the rational function field K(x(g) : g ∈ G) by K-automorphisms defined by g · x(h) = x(gh) for any g, h ∈ G. Denote by K(G) the fixed field K(x(g) : g ∈ G)G. Noether's problem then asks whether K(G) is rational over K. Let p be an odd prime and let G be a p-group of exponent pe. Assume also that (i) char K = p > 0, or (ii) char K ≠ p and K contains a primitive pe-th root of unity. In this paper we prove that K(G) is rational over K for the following two types of groups: (1) G is a finite p-group with an abelian normal subgroup H of index p such that H is a direct product of normal subgroups of G of the type Cpb × (Cp)c for some b, c with 1 ≤ b and 0 ≤ c; (2) G is any group of order p5 from the isoclinic families with numbers 1, 2, 3, 4, 8 and 9.