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.

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 Normal subgroups of finite multiply transitive permutation groups

1974 ◽  
Vol 53 ◽  
pp. 103-107 ◽  
Author(s):  
Eiichi Bannai
Keyword(s):  
Normal Subgroup ◽  
Permutation Group ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Normal Subgroups ◽  
Multiply Transitive

Wagner [5] and Ito [2] proved the following theorems respectively.THEOREM OF WAGNER. Let G be a triply transitive permutation group on a set Ω = {1,2, …, n}, and let n be odd and n > 4. If H is a normal subgroup (≠1) of G, then H is also triply transitive on Ω.

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 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.

scholarly journals Affine Primitive Groups and Semisymmetric Graphs

10.37236/2549 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Hua Han ◽  
Zaiping Lu
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Infinite Family ◽  
Permutation Groups ◽  
Primitive Groups ◽  
The Third ◽  
Semisymmetric Graphs

In this paper, we investigate semisymmetric graphs which arise from affine primitive permutation groups. We give a characterization of such graphs, and then construct an infinite family of semisymmetric graphs which contains the Gray graph as the third smallest member. Then, as a consequence, we obtain a factorization,of the complete bipartite graph $K_{p^{sp^t},p^{sp^t}}$ into connected semisymmetric graphs, where $p$ is an prime, $1\le t\le s$ with $s\ge2$ while $p=2$.

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