scholarly journals Normal subgroups in limit groups of prime index

2018 ◽  
Vol 21 (1) ◽  
pp. 83-100 ◽  
Author(s):  
Thomas S. Weigel ◽  
Jhoel S. Gutierrez
Keyword(s):  
Special Kind ◽  
Affirmative Answer ◽  
Limit Group ◽  
Limit Groups ◽  
Number Of Generators ◽  
Analogous Question ◽  
Minimum Number ◽  
Original Question ◽  
Rational Rank ◽  
Torsion Free

AbstractMotivated by their study of pro-plimit groups, D. H. Kochloukova and P. A. Zalesskii formulated in [15, Remark after Theorem 3.3] a question concerning the minimum number of generators{d(N)}of a normal subgroupNof prime indexpin a non-abelian limit groupG(see Question*). It is shown that the analogous question for the rational rank has an affirmative answer (see Theorem A). From this result one may conclude that the original question of Kochloukova and Zalesskii has an affirmative answer if the abelianization{G^{\mathrm{ab}}}ofGis torsion free and{d(G)=d(G^{\mathrm{ab}})}(see Corollary B), or ifGis a special kind of one-relator group (see Theorem D).

scholarly journals Quantitative residual properties of Γ-limit groups

2014 ◽  
Vol 24 (02) ◽  
pp. 207-231
Author(s):  
Brent B. Solie
Keyword(s):  
Hyperbolic Group ◽  
Finitely Generated ◽  
Limit Group ◽  
Limit Groups ◽  
Residual Properties

Let Γ be a fixed hyperbolic group. The Γ-limit groups of Sela are exactly the finitely generated, fully residually Γ groups. We introduce a new invariant of Γ-limit groups called Γ-discriminating complexity. We further show that the Γ-discriminating complexity of any Γ-limit group is asymptotically dominated by a polynomial.

On Automorphisms of Complete Algebras And The Isomorphism Problem for Modular Group Rings

1990 ◽  
Vol 42 (3) ◽  
pp. 383-394 ◽  
Author(s):  
Frank Röhl
Keyword(s):  
Group Ring ◽  
Modular Group ◽  
Isomorphism Problem ◽  
Nilpotency Class ◽  
Affirmative Answer ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Analogous Question

In [5], Roggenkamp and Scott gave an affirmative answer to the isomorphism problem for integral group rings of finite p-groups G and H, i.e. to the question whether ZG ⥲ ZH implies G ⥲ H (in this case, G is said to be characterized by its integral group ring). Progress on the analogous question with Z replaced by the field Fp of p elements has been very little during the last couple of years; and the most far reaching result in this area in a certain sense - due to Passi and Sehgal, see [8] - may be compared to the integral case, where the group G is of nilpotency class 2.

A Theorem on Pure Submodules

1960 ◽  
Vol 12 ◽  
pp. 483-487
Author(s):  
George Kolettis
Keyword(s):  
Free Abelian Group ◽  
Chain Condition ◽  
Affirmative Answer ◽  
Principal Ideal Ring ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Rank One ◽  
Pure Submodules ◽  
Torsion Free ◽  
Countable Case

In (1) Baer studied the following problem: If a torsion-free abelian group G is a direct sum of groups of rank one, is every direct summand of G also a direct sum of groups of rank one? For groups satisfying a certain chain condition, Baer gave a solution. Kulikov, in (3), supplied an affirmative answer, assuming only that G is countable. In a recent paper (2), Kaplansky settles the issue by reducing the general case to the countable case where Kulikov's solution is applicable. As usual, the result extends to modules over a principal ideal ring R (commutative with unit, no divisors of zero, every ideal principal).The object of this paper is to carry out a similar investigation for pure submodules, a somewhat larger class of submodules than the class of direct summands. We ask: if the torsion-free i?-module M is a direct sum of modules of rank one, is every pure submodule N of M also a direct sum of modules of rank one? Unlike the situation for direct summands, here the answer depends heavily on the ring R.

scholarly journals Commutator Length of Finitely Generated Linear Groups

2008 ◽  
Vol 2008 ◽  
pp. 1-5
Author(s):  
Mahboubeh Alizadeh Sanati
Keyword(s):  
Finite Group ◽  
Upper Bound ◽  
Quadratic Polynomial ◽  
Linear Groups ◽  
Number Of Generators ◽  
Derived Subgroup ◽  
Finite Linear Group ◽  
Minimum Number ◽  
Commutator Length ◽  
Finite Linear

The commutator length “” of a group is the least natural number such that every element of the derived subgroup of is a product of commutators. We give an upper bound for when is a -generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator length of a soluble-by-finite linear group over that depends only on and the degree of linearity. For such a group , we prove that is less than , where is the minimum number of generators of (upper) triangular subgroup of and is a quadratic polynomial in . Finally we show that if is a soluble-by-finite group of Prüffer rank then , where is a quadratic polynomial in .

The multi-region index of a knot

2020 ◽  
Vol 29 (04) ◽  
pp. 2050022
Author(s):  
Sarah Goodhill ◽  
Adam M. Lowrance ◽  
Valeria Munoz Gonzales ◽  
Jessica Rattray ◽  
Amelia Zeh
Keyword(s):  
Lower Bound ◽  
Crossing Number ◽  
Number Of Generators ◽  
Branched Cover ◽  
Minimum Number

Using region crossing changes, we define a new invariant called the multi-region index of a knot. We prove that the multi-region index of a knot is bounded from above by twice the crossing number of the knot. In addition, we show that the minimum number of generators of the first homology of the double branched cover of [Formula: see text] over the knot is strictly less than the multi-region index. Our proof of this lower bound uses Goeritz matrices.

scholarly journals Growth sequences of finite semigroups

1987 ◽  
Vol 43 (1) ◽  
pp. 16-20 ◽  
Author(s):  
James Wiegold ◽  
H. Lausch
Keyword(s):  
Real Number ◽  
Finite Semigroup ◽  
Piecewise Linear ◽  
The Other ◽  
Direct Power ◽  
Growth Sequence ◽  
Group Of Units ◽  
Number Of Generators ◽  
Finite Semigroups ◽  
Minimum Number

AbstractThe growth sequence of a finite semigroup S is the sequence {d(Sn)}, where Sn is the nth direct power of S and d stands for minimum generating number. When S has an identity, d(Sn) = d(Tn) + kn for all n, where T is the group of units and k is the minimum number of generators of S mod T. Thus d(Sn) is essentially known since d(Tn) is (see reference 4), and indeed d(Sn) is then eventually piecewise linear. On the other hand, if S has no identity, there exists a real number c > 1 such that d(Sn) ≥ cn for all n ≥ 2.

scholarly journals Pinning Decision in Interconnected Systems with Communication Disruptions under Multi-Agent Distributed Control Topology

2021 ◽  
Vol 2 (1) ◽  
pp. 18-36
Author(s):  
Samson S. Yu ◽  
Tat Kei Chau
Keyword(s):  
Decision Making ◽  
Transient Stability ◽  
Frequency Control ◽  
Automatic Generation ◽  
Distributed Generators ◽  
Number Of Generators ◽  
Minimum Number ◽  
Multi Agent ◽  
Active Communication ◽  
System Topology

In this study, we propose a decision-making strategy for pinning-based distributed multi-agent (PDMA) automatic generation control (AGC) in islanded microgrids against stochastic communication disruptions. The target microgrid is construed as a cyber-physical system, wherein the physical microgrid is modeled as an inverter-interfaced autonomous grid with detailed system dynamic formulation, and the communication network topology is regarded as a cyber-system independent of its physical connection. The primal goal of the proposed method is to decide the minimum number of generators to be pinned and their identities amongst all distributed generators (DGs). The pinning-decisions are made based on complex network theories using the genetic algorithm (GA), for the purpose of synchronizing and regulating the frequencies and voltages of all generator bus-bars in a PDMA control structure, i.e., without resorting to a central AGC agent. Thereafter, the mapping of cyber-system topology and the pinning decision is constructed using deep-learning (DL) technique, so that the pinning-decision can be made nearly instantly upon detecting a new cyber-system topology after stochastic communication disruptions. The proposed decision-making approach is verified using a 10-generator, 38-bus microgrid through time-domain simulation for transient stability analysis. Simulations show that the proposed pinning decision making method can achieve robust frequency control with minimum number of active communication channels.

Sign in / Sign up

Export Citation Format

Share Document