The Minimum Number of Generators of a Finite p -Group

1973 ◽  
Vol 5 (3) ◽  
pp. 288-290 ◽  
Author(s):  
Thomas J. Laffey
Keyword(s):  
Number Of Generators ◽  
Minimum Number

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.

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).

Growth sequences of 2-generator simple groups

1993 ◽  
Vol 123 (5) ◽  
pp. 839-855 ◽  
Author(s):  
V. N. Obraztsov
Keyword(s):  
Simple Groups ◽  
Direct Power ◽  
Number Of Generators ◽  
Minimum Number

SynopsisA study is made of the minimum number of generators of the n-th direct power of certain 2-generator groups.

scholarly journals Wirtinger numbers for virtual links

2019 ◽  
Vol 28 (14) ◽  
pp. 1950086
Author(s):  
Puttipong Pongtanapaisan
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Virtual Link ◽  
Link Diagram ◽  
Virtual Knots ◽  
Link Group ◽  
Number Of Generators ◽  
Virtual Links ◽  
Minimum Number ◽  
Bridge Number

The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number is algorithmically computable, it gives a more effective way to calculate an upper bound for the virtual bridge number from a virtual link diagram. As an application, we compute upper bounds for the virtual bridge numbers and the quandle counting invariants of virtual knots with 6 or fewer crossings. In particular, we found new examples of nontrivial virtual bridge number one knots, and by applying Satoh’s Tube map to these knots we can obtain nontrivial weakly superslice links.

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