Minimal presentations for groups of order 2n, n ≤ 6

1973 ◽  
Vol 15 (4) ◽  
pp. 461-469 ◽  
Author(s):  
T. W. Saga ◽  
J. W. Wamsley
Keyword(s):  
Generating Set ◽  
Image Position ◽  
Minimal Presentations ◽  
Minimal Generating Set

Let G be a finite 2-group having a minimal generating set {x1, …, xr} so that r = d (G) is an invariant of G. Suppose further that G has a presentation then.

Idempotents in partial transformation semigroups

1990 ◽  
Vol 116 (3-4) ◽  
pp. 359-366 ◽  
Author(s):  
G. U. Garba
Keyword(s):  
Real Number ◽  
Stirling Number ◽  
Partial Transformation ◽  
Transformation Semigroups ◽  
Generating Set ◽  
Idempotent Rank ◽  
Image Position ◽  
Minimal Generating Set

SynopsisAn element α of Pn, the semigroup of all partial transformations of {1,2,…, n}, is said to have projection characteristic (r, s), or to belong to the set [r, s], if dom α= r, im α = s. Let E be the set of all idempotents in Pn\[n, n] and E1, the set of those idempotents with projection characteristic (n, n − 1) or (n − 1, n − 1). For α in Pn\[n, n], we define a number g(α), called the gravity of α and closely related to the number denned in Howie [5] for full transformations, and we obtain the result thatLet d(α) be the defect of α, and for any real number x let [x] be the least integer m such that m ≧ x. Then by analogy with the results of Saito [9] we have thatα ϵ Ek(α) and α ∉ Ek(α)where k(α) = [g(α)/d(α)] or [g(α)/d(α)+ 1. Following Howie, Lusk and McFadden [6] we then explore connections between the defect and the gravity of α. Letting K(n, r) be the subsemigroup of Pn consisting of all elements of rank r or less, we prove a result, corresponding to that of Howie and McFadden [7] for total transformations, that the idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of K(n, r) is S(n + 1, r + 1), the Stirling number of the second kind.

scholarly journals Multiplicities, invariant subspaces and an additive formula

2021 ◽  
pp. 1-19
Author(s):  
Arup Chattopadhyay ◽  
Jaydeb Sarkar ◽  
Srijan Sarkar
Keyword(s):  
Hardy Space ◽  
Invariant Subspace ◽  
Dirichlet Space ◽  
Invariant Subspaces ◽  
Linear Operators ◽  
Bounded Linear ◽  
Generating Set ◽  
Image Position ◽  
Special Case ◽  
Minimal Generating Set

Abstract Let $T = (T_1, \ldots , T_n)$ be a commuting tuple of bounded linear operators on a Hilbert space $\mathcal{H}$ . The multiplicity of $T$ is the cardinality of a minimal generating set with respect to $T$ . In this paper, we establish an additive formula for multiplicities of a class of commuting tuples of operators. A special case of the main result states the following: Let $n \geq 2$ , and let $\mathcal{Q}_i$ , $i = 1, \ldots , n$ , be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in $\mathbb {C}$ . If $\mathcal{Q}_i^{\bot }$ , $i = 1, \ldots , n$ , is a zero-based shift invariant subspace, then the multiplicity of the joint $M_{\textbf {z}} = (M_{z_1}, \ldots , M_{z_n})$ -invariant subspace $(\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{\perp }$ of the Dirichlet space or the Hardy space over the unit polydisc in $\mathbb {C}^{n}$ is given by \[ \mbox{mult}_{M_{\textbf{{z}}}|_{ (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}}}} (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}} = \sum_{i=1}^{n} (\mbox{mult}_{M_z|_{\mathcal{Q}_i^{{\perp}}}} (\mathcal{Q}_i^{\bot})) = n. \] A similar result holds for the Bergman space over the unit polydisc.

scholarly journals The Fundamental Group of Balanced Simplicial Complexes and Posets

10.37236/73 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Steven Klee
Keyword(s):  
Fundamental Group ◽  
Upper Bound ◽  
Large Family ◽  
Simplicial Complexes ◽  
Generating Set ◽  
Minimal Generating Set

We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets.

The Fuzziness of a Minimal Generating Set of Fedorov Groups

2021 ◽  
Vol 66 (6) ◽  
pp. 913-919
Author(s):  
A. M. Banaru ◽  
V. R. Shiroky ◽  
D. A. Banaru
Keyword(s):  
Generating Set ◽  
Minimal Generating Set

The Number of Generators of a Linear p-Group

1972 ◽  
Vol 24 (5) ◽  
pp. 851-858 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Generating Set ◽  
Nilpotence Class ◽  
Number Of Generators ◽  
Minimal Generating Set

Let G be a finite p-group, having a faithful character χ of degree f. The object of this paper is to bound the number, d(G), of generators in a minimal generating set for G in terms of χ and in particular in terms of f. This problem was raised by D. M. Goldschmidt, and solved by him in the case that G has nilpotence class 2.

scholarly journals Triple-crossing projections, moves on knots and links and their minimal diagrams

2020 ◽  
Vol 29 (04) ◽  
pp. 2050015 ◽  
Author(s):  
Michał Jabłonowski ◽  
Łukasz Trojanowski
Keyword(s):  
Triple Point ◽  
Crossing Number ◽  
Systematic Method ◽  
Generating Set ◽  
New Type ◽  
Knots And Links ◽  
Minimal Generating Set

In this paper, we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By introducing a new type of diagrammatic move, we reduce the number of generating moves on triple-crossing diagrams, and derive a minimal generating set of moves connecting triple-crossing diagrams of the same knot.

Generalization of the basis property on finite groups

2020 ◽  
pp. 2150111
Author(s):  
Ibrahim Al-Dayel ◽  
Ahmad Al Khalaf
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Frobenius Group ◽  
Sufficient Conditions ◽  
Basis Property ◽  
Generating Set ◽  
Equivalent Basis ◽  
Necessary And Sufficient ◽  
Special Case ◽  
Minimal Generating Set

A group [Formula: see text] has the Basis Property if every subgroup [Formula: see text] of [Formula: see text] has an equivalent basis (minimal generating set). We studied a special case of the finite group with the Basis Property, when [Formula: see text]-group [Formula: see text] is an abelian group. We found the necessary and sufficient conditions on an abelian [Formula: see text]-group [Formula: see text] of [Formula: see text] with the Basis Property to be kernel of Frobenius group.

scholarly journals The minimal generating sets of of size four

2013 ◽  
Vol 16 ◽  
pp. 419-423 ◽  
Author(s):  
Sebastian Jambor
Keyword(s):  
Generating Set ◽  
Generating Sets ◽  
Minimal Generating Set

AbstractWe show that there are only finitely many primes $p$ such that $\mathrm{PSL} (2, p)$ has a minimal generating set of size four.Supplementary materials are available with this article.

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