scholarly journals ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

2021 ◽  
pp. 1-40
Author(s):  
NICK GILL ◽  
BIANCA LODÀ ◽  
PABLO SPIGA
Keyword(s):  
Permutation Group ◽  
Model Theory ◽  
Independent Set ◽  
Maximum Size ◽  
Proper Subset ◽  
Permutation Groups ◽  
Finite Permutation Group ◽  
Pointwise Stabilizer ◽  
Relational Complexity

Abstract Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

scholarly journals On multiply transitive permutation groups

1978 ◽  
Vol 26 (1) ◽  
pp. 57-58
Author(s):  
G. P. Monro ◽  
D. E. Taylor
Keyword(s):  
Permutation Group ◽  
Subject Classification ◽  
Permutation Groups ◽  
Combinatorial Proof ◽  
Bell Numbers ◽  
Finite Permutation Group ◽  
Multiply Transitive

AbstractWe present a direct combinatorial proof of the characterization of the degree of transivity of a finite permutation group in terms of the Bell numbers.Subject classification (Amer. Math.Soc. (MOS) 1970): 20 B 20.

On the Group of a Directed Graph

1966 ◽  
Vol 18 ◽  
pp. 211-220 ◽  
Author(s):  
Robert L. Hemminger
Keyword(s):  
Finite Group ◽  
Permutation Group ◽  
Directed Graph ◽  
Directed Graphs ◽  
Symmetric Case ◽  
Related Question ◽  
Permutation Groups ◽  
Symmetric Graph ◽  
Finite Permutation Group ◽  
Symmetric Graphs

In 1938, Frucht (2) proved that for any given finite group G there exists a finite symmetric graph X such that G(X) is abstractly isomorphic to G. Since G(X) is a permutation group, it is natural to ask the following related question : If P is a given finite permutation group, does there exist a symmetric (and more generally a directed) graph X such that G(X) and P are isomorphic (see Convention below) as permutation groups? The answer for the symmetric case is negative as seen in (3) and more recently in (1). It is the purpose of this paper to deal with this problem further, especially in the directed case. In §3, we supplement Kagno's results (3, pp. 516-520) for symmetric graphs by giving the corresponding results for directed graphs.

Magic letter groups

2007 ◽  
Vol 91 (522) ◽  
pp. 493-499
Author(s):  
Mike Pearson ◽  
Ian Short
Keyword(s):  
Permutation Group ◽  
Isomorphism Type ◽  
Permutation Groups ◽  
Natural Manner ◽  
Finite Permutation Group

Certain numeric puzzles, known as ‘magic letters’, each have a finite permutation group associated with them in a natural manner. We describe how the isomorphism type of these permutation groups relates to the structure of the magic letters.

scholarly journals Quotients of permutation groups

1998 ◽  
Vol 57 (3) ◽  
pp. 493-495
Author(s):  
John Cossey
Keyword(s):  
Normal Subgroup ◽  
Permutation Group ◽  
Faithful Representation ◽  
Permutation Representation ◽  
Permutation Groups ◽  
Fitting Subgroup ◽  
Finite Permutation Group ◽  
Special Cases

If G is a finite permutation group of degree d and N is a normal subgroup of G, Derek Holt has given conditions which show that in some important special cases the least degree of a faithful permutation representation of the quotient G/N will be no larger than d. His conditions do not apply in all cases of interest and he remarks that it would be interesting to know if G/F(G) has a faithful representation of degree no larger than d (where F(G) is the Fitting subgroup of G). We prove in this note that this is the case.

scholarly journals On permutation groups with constant movement

2003 ◽  
Vol 74 (2) ◽  
pp. 287-294
Author(s):  
Mehdi Alaeiyan(Khayaty)
Keyword(s):  
Fixed Point ◽  
Permutation Group ◽  
Maximum Size ◽  
Permutation Groups ◽  
Identity Elements

AbstractLet G be a permutation group on a set Ω with no fixed point in Ω. If for each subset Г of Ω the size |Гg - Г| is bounded, for g ∈ G, we define the movement of g as the max|Гg − Г| over all subsets Г of Ω. In particular, if all non-identity elements of G have the same movement, then we say that G has constant movement. In this paper we will first give some families of groups with constant movement. We then classify all transitive permutation groups with a given constant movement m on a set of maximum size.

scholarly journals Base sizes of primitive permutation groups

2021 ◽  
Author(s):  
Mariapia Moscatiello ◽  
Colva M. Roney-Dougal
Keyword(s):  
Permutation Group ◽  
Wreath Product ◽  
Permutation Groups ◽  
Minimal Size ◽  
Mathieu Group ◽  
Natural Action ◽  
Primitive Groups ◽  
Product Action ◽  
Pointwise Stabilizer

AbstractLet G be a permutation group, acting on a set $$\varOmega $$ Ω of size n. A subset $${\mathcal {B}}$$ B of $$\varOmega $$ Ω is a base for G if the pointwise stabilizer $$G_{({\mathcal {B}})}$$ G ( B ) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of $$\mathrm {Sym}(n)$$ Sym ( n ) is large base if there exist integers m and $$r \ge 1$$ r ≥ 1 such that $${{\,\mathrm{Alt}\,}}(m)^r \unlhd G \le {{\,\mathrm{Sym}\,}}(m)\wr {{\,\mathrm{Sym}\,}}(r)$$ Alt ( m ) r ⊴ G ≤ Sym ( m ) ≀ Sym ( r ) , where the action of $${{\,\mathrm{Sym}\,}}(m)$$ Sym ( m ) is on k-element subsets of $$\{1,\dots ,m\}$$ { 1 , ⋯ , m } and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group $$\mathrm {M}_{24}$$ M 24 in its natural action on 24 points, or $$b(G)\le \lceil \log n\rceil +1$$ b ( G ) ≤ ⌈ log n ⌉ + 1 . Furthermore, we show that there are infinitely many primitive groups G that are not large base for which $$b(G) > \log n + 1$$ b ( G ) > log n + 1 , so our bound is optimal.

scholarly journals Characterization of micro-RNA in women with different ovarian reserve

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Masood Abu-Halima ◽  
Lea Simone Becker ◽  
Basim M. Ayesh ◽  
Simona Lucia Baus ◽  
Amer Hamza ◽  
...  
Keyword(s):  
Ovarian Reserve ◽  
Independent Set ◽  
Micro Rna ◽  
Mirna Array ◽  
Reverse Transcription Quantitative Pcr ◽  
Diagnostic Approaches ◽  
Auc Value ◽  
Mirna Abundance

AbstractWomen undergoing infertility treatment are routinely subjected to one or more tests of ovarian reserve. Therefore, an adequate assessment of the ovarian reserve is necessary for the treatment. In this study, we aimed to characterize the potential role of microRNAs (miRNAs) as biomarkers for women with different ovarian reserves. A total of 159 women were recruited in the study and classified according to their anti-Müllerian hormone (AMH) level into three groups: (1) low ovarian reserve (LAMH, n = 39), (2) normal ovarian reserve (NAMH, n = 80), and (3) high ovarian reserve (HAMH, n = 40). SurePrint Human miRNA array screening and reverse transcription-quantitative PCR (RT-qPCR) were respectively employed to screen and validate the miRNA abundance level in the three tested groups. Compared with NAMH, the abundance level of 34 and 98 miRNAs was found to be significantly altered in LAMH and HAMH, respectively. The abundance level of miRNAs was further validated by RT-qPCR in both, the screening samples as well as in an independent set of validation samples. The abundance levels of the validated miRNAs were significantly correlated with the AMH level. The best AUC value for the prediction of the increase and decrease in the AMH level was obtained for the miR-100-5p and miR-21-5p, respectively. The level of miRNAs abundance correlates with the level of AMH, which may serve as a tool for identifying women with a different ovarian reserve and may help to lay the ground for the development of novel diagnostic approaches.

scholarly journals Transversals in permutation groups and factorisations of complete graphs

2002 ◽  
Vol 65 (2) ◽  
pp. 277-288 ◽  
Author(s):  
Gil Kaplan ◽  
Arieh Lev
Keyword(s):  
Permutation Group ◽  
Directed Graph ◽  
Sufficient Conditions ◽  
Combinatorial Problems ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Complete Graphs ◽  
Frobenius Theorem ◽  
Disjoint Cycles ◽  
Finite Set

Let G be a transitive permutation group acting on a finite set of order n. We discuss certain types of transversals for a point stabiliser A in G: free transversals and global transversals. We give sufficient conditions for the existence of such transversals, and show the connection between these transversals and combinatorial problems of decomposing the complete directed graph into edge disjoint cycles. In particular, we classify all the inner-transitive Oberwolfach factorisations of the complete directed graph. We mention also a connection to Frobenius theorem.

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