scholarly journals A sharply 2-transitive group without a non-trivial abelian normal subgroup

2017 ◽  
Vol 19 (10) ◽  
pp. 2895-2910 ◽  
Author(s):  
Eliyahu Rips ◽  
Yoav Segev ◽  
Katrin Tent
Keyword(s):  
Normal Subgroup ◽  
Transitive Group ◽  
Abelian Normal Subgroup

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

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Yuting Hu ◽  
Yidun Wan
Keyword(s):  
Normal Subgroup ◽  
Gauge Group ◽  
The Other ◽  
Quantum Double ◽  
Duality Transformation ◽  
Abelian Normal Subgroup

Abstract 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.

Notes on Splitting Extensions of Groups

1968 ◽  
Vol 11 (3) ◽  
pp. 371-374 ◽  
Author(s):  
C.Y. Tang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Commutator Subgroup ◽  
Similar Type ◽  
Frattini Subgroup ◽  
Abelian Normal Subgroup ◽  
A Finite Group

In [1] Gaschütz has shown that a finite group G splits over an abelian normal subgroup N if its Frattini subgroup ϕ(G) intersects N trivially. When N is a non-abelian nilpotent normal subgroup of G the condition ϕ(G)∩ N = 1 cannot be satisfied: for if N is non-abelian then the commutator subgroup C(N) of N is non-trivial. Now N is nilpotent, whence 1 ≠ C(N)⊂ϕ(N). Since G is a finite group, therefore, by (3, theorem 7.3.17) ϕ⊂ϕ(G). It follows that ϕ(G) ∩ N ≠ 1. Thus the condition ϕ(G) ∩ N = 1 must be modified. In §1 we shall derive some similar type of conditions for G to split over N when the restriction of N being an abelian normal subgroup is removed. In § 2 we shall give a characterization of splitting extensions of N in which every subgroup splits over its intersection with N.

Degrees of faithful irreducible representations of metabelian groups

2021 ◽  
pp. 2250181
Author(s):  
Rahul Dattatraya Kitture ◽  
Soham Swadhin Pradhan
Keyword(s):  
Normal Subgroup ◽  
Finite Groups ◽  
Positive Characteristic ◽  
Number Fields ◽  
Irreducible Representations ◽  
Modular Representations ◽  
Metacyclic Group ◽  
Metabelian Groups ◽  
Abelian Normal Subgroup ◽  
Field Of Positive Characteristic

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.

Critical maximal subgroups and conjugacy of supplements in finite soluble groups

2018 ◽  
Vol 21 (1) ◽  
pp. 45-63
Author(s):  
Barbara Baumeister ◽  
Gil Kaplan
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Maximal Subgroups ◽  
Abelian Normal Subgroup ◽  
Soluble Groups ◽  
A Finite Group

AbstractLetGbe 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.

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

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Albert Garreta ◽  
Leire Legarreta ◽  
Alexei Miasnikov ◽  
Denis Ovchinnikov
Keyword(s):  
Normal Subgroup ◽  
Full Rank ◽  
Direct Decomposition ◽  
Metabelian Groups ◽  
Abelian Normal Subgroup ◽  
Diophantine Problem ◽  
Diophantine Problems ◽  
Finitely Presented ◽  
Finite Presentations

AbstractWe 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.

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