scholarly journals THE CONTRIBUTION OF L. G. KOVÁCS TO THE THEORY OF PERMUTATION GROUPS

2015 ◽  
Vol 102 (1) ◽  
pp. 20-33
Author(s):  
CHERYL E. PRAEGER ◽  
CSABA SCHNEIDER
Keyword(s):  
Permutation Groups ◽  
Primitive Groups

The work of L. G. (Laci) Kovács (1936–2013) gave us a deeper understanding of permutation groups, especially in the O’Nan–Scott theory of primitive groups. We review his contribution to this field.

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 $p=2$.

scholarly journals Derangements in cosets of primitive permutation groups

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Andrei Pavelescu
Keyword(s):  
Fixed Point ◽  
Finite Fields ◽  
Permutation Groups ◽  
Normal Subgroups ◽  
Primitive Groups ◽  
Branched Coverings ◽  
Affine Groups ◽  
Nonabelian Subgroup ◽  
Projective Curves ◽  
Curves Over Finite Fields

AbstractMotivated by questions arising in connection with branched coverings of connected smooth projective curves over finite fields, we study the proportion of fixed-point free elements (derangements) in cosets of normal subgroups of primitive permutations groups. Using the Aschbacher–O'Nan–Scott Theorem for primitive groups to partition the problem, we provide complete answers for affine groups and groups which contain a regular normal nonabelian subgroup.

scholarly journals Bounds and quotient actions of innately transitive groups

2005 ◽  
Vol 79 (1) ◽  
pp. 95-112 ◽  
Author(s):  
John Bamberg
Keyword(s):  
Permutation Groups ◽  
Primitive Groups ◽  
Transitive Groups ◽  
Parameter Bounds

AbstractFinite innately transitive permutation groups include all finite quasiprimitive and primitive permutation groups. In this paper, results in the theory of quasiprimitive and primitive groups are generalised to innately transitive groups, and in particular, we extend results of Praeger and Shalev. Thus we show that innately transitive groups possess parameter bounds which are similar to those for primitive groups. We also classify the innately transitive types of quotient actions of innately transitive groups.

scholarly journals Invariance groups of finite functions and orbit equivalence of permutation groups

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Eszter K. Horváth ◽  
Géza Makay ◽  
Reinhard Pöschel ◽  
Tamás Waldhauser
Keyword(s):  
Symmetric Group ◽  
Partial Answer ◽  
Permutation Groups ◽  
Computational Results ◽  
Orbit Equivalence ◽  
Alternating Groups ◽  
Primitive Groups ◽  
Complete Answer ◽  
The Difference ◽  
Invariance Groups

AbstractWhich subgroups of the symmetric group Sn arise as invariance groups of n-variable functions defined on a k-element domain? It appears that the higher the difference n-k, the more difficult it is to answer this question. For k ≤ n, the answer is easy: all subgroups of Sn are invariance groups. We give a complete answer in the cases k = n-1 and k = n-2, and we also give a partial answer in the general case: we describe invariance groups when n is much larger than n-k. The proof utilizes Galois connections and the corresponding closure operators on Sn, which turn out to provide a generalization of orbit equivalence of permutation groups. We also present some computational results, which show that all primitive groups except for the alternating groups arise as invariance groups of functions defined on a three-element domain.

scholarly journals A Note on Primitive Permutation Groups of Prime Power Degree

2015 ◽  
Vol 2015 ◽  
pp. 1-4
Author(s):  
Qian Cai ◽  
Hua Zhang
Keyword(s):  
Prime Power ◽  
Permutation Groups ◽  
Simple Type ◽  
Present Stage ◽  
Primitive Groups ◽  
Short Survey ◽  
Product Action ◽  
Action Type ◽  
Affine Type

Primitive permutation groups of prime power degree are known to be affine type, almost simple type, and product action type. At the present stage finding an explicit classification of primitive groups of affine type seems untractable, while the product action type can usually be reduced to almost simple type. In this paper, we present a short survey of the development of primitive groups of prime power degree, together with a brief description on such groups.

Bounds for finite semiprimitive permutation groups: order, base size, and minimal degree

2020 ◽  
Vol 63 (4) ◽  
pp. 1071-1091
Author(s):  
Luke Morgan ◽  
Cheryl E. Praeger ◽  
Kyle Rosa
Keyword(s):  
Fixed Point ◽  
Normal Subgroup ◽  
Symmetric Group ◽  
Minimal Normal Subgroup ◽  
Structural Information ◽  
Minimal Degree ◽  
Permutation Groups ◽  
Primitive Groups ◽  
Base Size ◽  
Abstract Structure

AbstractIn this paper, we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. These groups have recently been investigated in terms of their abstract structure, in a similar way to the O'Nan–Scott Theorem for primitive groups. Our goal here is to explore aspects of such groups which may be useful in place of precise structural information. We give bounds on the order, base size, minimal degree, fixed point ratio, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we study the structure of a finite semiprimitive group that induces the alternating or symmetric group on the set of orbits of an intransitive minimal normal subgroup.

ON MINIMAL SUBGROUP COVERINGS OF SOME AFFINE PRIMITIVE PERMUTATION GROUPS

2010 ◽  
Vol 09 (06) ◽  
pp. 985-987 ◽  
Author(s):  
SEYYED MAJID JAFARIAN AMIRI
Keyword(s):  
Open Problem ◽  
Solvable Groups ◽  
Permutation Groups ◽  
Minimal Subgroup ◽  
Primitive Groups

A cover for a group G is a collection of proper subgroups of G whose union is G. Cohn defined σ(G) to be the least integer m such that G is the union of m proper subgroups. Determining σ is an open problem for most non-solvable groups. We obtain σ(G) for some affine primitive groups.

scholarly journals Coprime subdegrees of twisted wreath permutation groups

2019 ◽  
Vol 62 (4) ◽  
pp. 1137-1162
Author(s):  
Alexander Y. Chua ◽  
Michael Giudici ◽  
Luke Morgan
Keyword(s):  
Simple Group ◽  
Permutation Group ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Primitive Groups ◽  
Group A ◽  
Full Classification

AbstractDolfi, Guralnick, Praeger and Spiga asked whether there exist infinitely many primitive groups of twisted wreath type with non-trivial coprime subdegrees. Here, we settle this question in the affirmative. We construct infinite families of primitive twisted wreath permutation groups with non-trivial coprime subdegrees. In particular, we define a primitive twisted wreath group G(m, q) constructed from the non-abelian simple group PSL(2, q) and a primitive permutation group of diagonal type with socle PSL(2, q)m, and determine many subdegrees for this group. A consequence is that we determine all values of m and q for which G(m, q) has non-trivial coprime subdegrees. In the case where m = 2 and $q\notin \{7,11,29\}$, we obtain a full classification of all pairs of non-trivial coprime subdegrees.

The primitive permutation groups of degree less than 1000

1988 ◽  
Vol 103 (2) ◽  
pp. 213-238 ◽  
Author(s):  
John D. Dixon ◽  
Brian Mortimer
Keyword(s):  
Finite Groups ◽  
Primitive Group ◽  
Simple Groups ◽  
Permutation Groups ◽  
Finite Simple Groups ◽  
Primitive Groups ◽  
Concrete Form

Our object is to describe all of the-primitive permutation groups of degree less than 1000 together with some of their significant properties. We think that such a list is of interest in illustrating in concrete form the kinds of primitive groups which arise, in suggesting conjectures about primitive groups, and in settling small exceptional cases which often occur in proofs of theorems about permutation groups. The range that we consider is large enough to allow examples of most of the types of primitive group to appear. Earlier lists (of varying completeness and accuracy) of primitive groups of degree d have been published by: C. Jordan (1872) [21] ford≤ 17, by W. Burnside (1897) [5] ford≤ 8, by Manning (1929) [34–38] ford≤ 15, by C. C. Sims (1970) [45] ford≤ 20, and by B. A. Pogorelev (1980) [42] ford≤ 50. Unpublished lists have also been prepared by C. C. Sims ford≤ 50 and by Mizutani[41] ford≤ 48. Using the classification of finite simple groups which was completed in 1981 we have been able to extend the list much further. Our task has been greatly simplified by the detailed information about many finite simple groups which is available in theAtlas of Finite Groupswhich we will refer to as theAtlas[8].

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.

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