scholarly journals The minimal base size for a $p$-solvable linear group

2015 ◽  
Vol 144 (8) ◽  
pp. 3231-3242 ◽  
Author(s):  
Zoltán Halasi ◽  
Attila Maróti
Keyword(s):  
Linear Group ◽  
Minimal Base ◽  
Base Size ◽  
Solvable Linear Group

scholarly journals Random bases for coprime linear groups

2020 ◽  
Vol 23 (1) ◽  
pp. 133-157
Author(s):  
Hülya Duyan ◽  
Zoltán Halasi ◽  
Károly Podoski
Keyword(s):  
Group Theory ◽  
Permutation Group ◽  
Linear Group ◽  
Additional Assumption ◽  
Probabilistic Method ◽  
Personal Communication ◽  
Linear Groups ◽  
Minimal Base ◽  
Base Size ◽  
Almost All

AbstractThe minimal base size {b(G)} for a permutation group G is a widely studied topic in permutation group theory. Z. Halasi and K. Podoski [Every coprime linear group admits a base of size two, Trans. Amer. Math. Soc. 368 2016, 8, 5857–5887] proved that {b(G)\leq 2} for coprime linear groups. Motivated by this result and the probabilistic method used by T. Burness, M. W. Liebeck and A. Shalev, it was asked by L. Pyber [Personal communication, Bielefeld, 2017] whether or not, for coprime linear groups {G\leq GL(V)}, there exists a constant c such that the probability that a random c-tuple is a base for G tends to 1 as {\lvert V\rvert\to\infty}. While the answer to this question is negative in general, it is positive under the additional assumption that G is primitive as a linear group. In this paper, we show that almost all 11-tuples are bases for coprime primitive linear groups.

Normal p-subgroups of solvable linear groups

1967 ◽  
Vol 7 (4) ◽  
pp. 545-551 ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Linear Group ◽  
Linear Groups ◽  
Complex Numbers ◽  
Solvable Linear Group ◽  
Elegant Proof

In his paper [8], N. Itô gives an elegant proof that the Sylow p-group of a finite solvable linear group of degree n over the field of complex numbers is necessarily normal if p > n+1. Moreover he shows that this bound on p is the best possible when p is a Fermat prime (i.e. a prime of the form 2sk + 1) but that the bound may be improved to p > n when p is not a Fermat prime.

scholarly journals On the base size for the symmetric group acting on subsets

2012 ◽  
Vol 49 (4) ◽  
pp. 492-500 ◽  
Author(s):  
Zoltán Halasi
Keyword(s):  
Symmetric Group ◽  
Universal Constant ◽  
Natural Numbers ◽  
General Process ◽  
Minimal Base ◽  
Base Size ◽  
Natural Way

Let k, n be natural numbers with k ≦ n/2 and let Xn,k denote the set of k-element subsets of {1, 2, … n}. The symmetric group Sn acts in a natural way on the set Xn,k. Motivated by a question of Robert Guralnick, we investigate the size of a minimal base for this action. We give constructions providing a minimal base if n = 2k or if n ≧ k2. We also describe a general process providing a base of size at most c times bigger than the size of a minimal base for some universal constant c

On generic complexity of the subset sum problem in monoids and groups of integer matrix of order two

2020 ◽  
Vol 25 (4) ◽  
pp. 10-15
Author(s):  
Alexander Nikolaevich Rybalov
Keyword(s):  
Linear Group ◽  
Modular Group ◽  
Special Linear Group ◽  
Second Order ◽  
Integer Matrix ◽  
Subset Sum ◽  
Algorithmic Problems ◽  
Almost All ◽  
Subset Sum Problems ◽  
Generic Complexity

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

Association Study of Fecundity Gene BMPR1B with Prolificacy in Surti Goats under Farm and Field Condition

2019 ◽  
Vol 14 (04) ◽  
Author(s):  
N. S. Dangar ◽  
G. M. Pandya ◽  
U. V. Ramani ◽  
Y. D. Padheriya ◽  
T. Sangma ◽  
...  
Keyword(s):  
Litter Size ◽  
Transforming Growth Factor Beta ◽  
Transforming Growth Factor ◽  
Pcr Primers ◽  
Dual Purpose ◽  
Protein Receptor ◽  
Morphogenetic Protein ◽  
Pcr Rflp ◽  
Base Size ◽  
Mutation Information

The Surti is a dual purpose goat breed of Gujarat. The bone morphogenetic protein receptor type 1B (BMPR1B) gene of transforming growth factor beta (TGF-β) superfamily ligands is playing a role in ovulation as well as litter size. Mutation in Exon-6 region of BMPR1B gene with base size 190 bp reported increasing litter size. Based on the known mutation information in goat and sheep, PCR primers were designed to screen polymorphism in total 100 Surti goats, 50 Surti goats from University Farm, Navsari and 50 Surti goats from field units of Southern part of Gujarat. During PCR-RFLP study no polymorphic sites were found for Exon-6 region of BMPR1B on Surti goats. Moreover, the twinning rate was 10% in first parity and higher in subsequent second (62.5%) and third (76.8%) parties.

scholarly journals Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds

2021 ◽  
Author(s):  
Adrien Laurent ◽  
Gilles Vilmart
Keyword(s):  
Invariant Measure ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Linear Group ◽  
Langevin Equation ◽  
Numerical Experiments ◽  
Special Linear Group ◽  
Order Conditions ◽  
High Order ◽  
Differential Equations On Manifolds

AbstractWe derive a new methodology for the construction of high-order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge–Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.

Representations induced from the Zelevinsky segment and discrete series in the half-integral case

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ivan Matić
Keyword(s):  
Linear Group ◽  
Local Field ◽  
General Linear Group ◽  
Discrete Series ◽  
Series Representation ◽  
General Linear ◽  
Composition Series ◽  
Induced Representations ◽  
Discrete Series Representation

AbstractLet {G_{n}} denote either the group {\mathrm{SO}(2n+1,F)} or {\mathrm{Sp}(2n,F)} over a non-archimedean local field of characteristic different than two. We study parabolically induced representations of the form {\langle\Delta\rangle\rtimes\sigma}, where {\langle\Delta\rangle} denotes the Zelevinsky segment representation of the general linear group attached to the segment Δ, and σ denotes a discrete series representation of {G_{n}}. We determine the composition series of {\langle\Delta\rangle\rtimes\sigma} in the case when {\Delta=[\nu^{a}\rho,\nu^{b}\rho]} where a is half-integral.

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