Supplements of bounded permutation groups

1998 ◽  
Vol 63 (1) ◽  
pp. 89-102 ◽  
Author(s):  
Stephen Bigelow
Keyword(s):  
Symmetric Group ◽  
Permutation Group ◽  
Permutation Groups ◽  
Cardinal Arithmetic

AbstractLet λ ≤ κ be infinite cardinals and let Ω be a set of cardinality κ. The bounded permutation group Bλ(Ω), or simply Bλ, is the group consisting of all permutations of Ω which move fewer than λ points in Ω. We say that a permutation group G acting on Ω is a supplement of Bλ if BλG is the full symmetric group on Ω.In [7], Macpherson and Neumann claimed to have classified all supplements of bounded permutation groups. Specifically, they claimed to have proved that a group G acting on the set Ω is a supplement of Bλ if and only if there exists Δ ⊂ Ω with ∣Δ∣ < λ such that the setwise stabiliser G{Δ} acts as the full symmetric group on Ω ∖ Δ. However I have found a mistake in their proof. The aim of this paper is to examine conditions under which Macpherson and Neumann's claim holds, as well as conditions under which a counterexample can be constructed. In the process we will discover surprising links with cardinal arithmetic and Shelah's recently developed pcf theory.

On linear p-groups

1964 ◽  
Vol 4 (2) ◽  
pp. 174-178 ◽  
Author(s):  
W. J. Wong
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Permutation Group ◽  
Faithful Representation ◽  
Permutation Groups ◽  
Derived Length ◽  
A Finite Group ◽  
Rational Integral

A quasi-permutation group of degree n was defined in [3] to be a finite group with a faithful representation of degree n whose character has only non-negative rational integral values. If G is such a group, then the following simple properties of permutation groups of degree n were proved to hold also for G:(i) the order of G is a divisor of the order of the symmetric group Sn of degree n; and (ii) if G is a p-group and n < p2, then G has exponent at most p and derived length at most 1 (i.e. G is elementary Abelian).

scholarly journals On Multiple Transitivity of Permutation Groups

1961 ◽  
Vol 18 ◽  
pp. 93-109 ◽  
Author(s):  
Tosiro Tsuzuku
Keyword(s):  
Symmetric Group ◽  
Permutation Group ◽  
Irreducible Character ◽  
Symmetric Groups ◽  
Transitive Group ◽  
Permutation Groups ◽  
Irreducible Characters ◽  
Interesting Theorem

It is well known that a doubly transitive group has an irreducible character χ1 such that χ1(R) = α(R) − 1 for any element R of and a quadruply transitive group has irreducible characters χ3 and χ3 such that χ2(R) = where α(R) and β(R) are respectively the numbers of one cycles and two cycles contained in R. G. Frobenius was led to this fact in the connection with characters of the symmetric groups and he proved the following interesting theorem: if a permutation group of degree n is t-ply transitive, then any irreducible character of the symmetric group of degree n with dimension at most equal to is an irreducible character of .

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 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 A Character Condition for Quadruple Transitivity

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
S. Aldhafeeri ◽  
R. T. Curtis
Keyword(s):  
Symmetric Group ◽  
Permutation Group ◽  
Irreducible Character

Let be a permutation group of degree viewed as a subgroup of the symmetric group . We show that if the irreducible character of corresponding to the partition of into subsets of sizes and 2, that is, to say the character often denoted by , remains irreducible when restricted to , then = 4, 5 or 9 and , A5, or PΣL2(8), respectively, or is 4-transitive.

scholarly journals Enumerating Finite Racks, Quandles and Kei

10.37236/3262 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Simon R. Blackburn
Keyword(s):  
Permutation Group ◽  
Lower Bounds ◽  
Binary Operation ◽  
Independent Interest ◽  
Upper And Lower Bounds ◽  
Permutation Groups ◽  
Isomorphism Classes ◽  
Operator Group ◽  
The Relationship

A rack of order $n$ is a binary operation $\vartriangleright$ on a set $X$ of cardinality $n$, such that right multiplication is an automorphism. More precisely, $(X,\vartriangleright)$ is a rack provided that the map $x\mapsto x\vartriangleright y$ is a bijection for all $y\in X$, and $(x\vartriangleright y)\vartriangleright z=(x\vartriangleright z)\vartriangleright (y\vartriangleright z)$ for all $x,y,z\in X$.The paper provides upper and lower bounds of the form $2^{cn^2}$ on the number of isomorphism classes of racks of order $n$. Similar results on the number of isomorphism classes of quandles and kei are obtained. The results of the paper are established by first showing how an arbitrary rack is related to its operator group (the permutation group on $X$ generated by the maps $x\mapsto x\vartriangleright y$ for $y\in Y$), and then applying some of the theory of permutation groups. The relationship between a rack and its operator group extends results of Joyce and of Ryder; this relationship might be of independent interest.

scholarly journals Groups fixing graphas in switching classes

1982 ◽  
Vol 33 (1) ◽  
pp. 76-85
Author(s):  
Martin W. Liebeck
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Permutation Group ◽  
Abelian Groups ◽  
Finite Set ◽  
A Finite Group ◽  
Switching Class

AbstractA permutation group G on a finite set Ω is always exposable if whenever G stabilises a switching class of graphs on Ω, G fixes a graph in the switching class. Here we consider the problem: given a finite group G, which permutation representations of G are always exposable? We present solutions to the problem for (i) 2-generator abelian groups, (ii) all abelian groups in semiregular representations. (iii) generalised quaternion groups and (iv) some representations of the symmetric group Sn.

Permutation groups and orbits on the power set

2019 ◽  
Vol 19 (12) ◽  
pp. 2150005
Author(s):  
Yong Yang
Keyword(s):  
Permutation Group ◽  
Permutation Groups ◽  
Power Set ◽  
Composition Factors

Let [Formula: see text] be a permutation group of degree [Formula: see text] and let [Formula: see text] denote the number of set-orbits of [Formula: see text]. We determine [Formula: see text] over all groups [Formula: see text] that satisfy certain restrictions on composition factors.

scholarly journals EMBEDDING PERMUTATION GROUPS INTO WREATH PRODUCTS IN PRODUCT ACTION

2012 ◽  
Vol 92 (1) ◽  
pp. 127-136 ◽  
Author(s):  
CHERYL E. PRAEGER ◽  
CSABA SCHNEIDER
Keyword(s):  
Permutation Group ◽  
Wreath Product ◽  
Automorphism Groups ◽  
Wreath Products ◽  
Error Correcting Codes ◽  
Permutation Groups ◽  
Graph Products ◽  
Product Action ◽  
Hamming Graphs

AbstractWe consider the wreath product of two permutation groups G≤Sym Γ and H≤Sym Δ as a permutation group acting on the set Π of functions from Δ to Γ. Such groups play an important role in the O’Nan–Scott theory of permutation groups and they also arise as automorphism groups of graph products and codes. Let X be a subgroup of Sym Γ≀Sym Δ. Our main result is that, in a suitable conjugate of X, the subgroup of SymΓ induced by a stabiliser of a coordinate δ∈Δ only depends on the orbit of δ under the induced action of X on Δ. Hence, if X is transitive on Δ, then X can be embedded into the wreath product of the permutation group induced by the stabiliser Xδ on Γ and the permutation group induced by X on Δ. We use this result to describe the case where X is intransitive on Δ and offer an application to error-correcting codes in Hamming graphs.

scholarly journals Bounds on finite quasiprimitive permutation groups

2001 ◽  
Vol 71 (2) ◽  
pp. 243-258 ◽  
Author(s):  
Cheryl E. Praeger ◽  
Aner Shalev
Keyword(s):  
Normal Subgroup ◽  
Permutation Group ◽  
Minimum Degree ◽  
Permutation Groups ◽  
Primitive Permutation Group ◽  
Base Size ◽  
Nontrivial Normal Subgroup

AbstractA permutation group is said to be quasiprimitive if every nontrivial normal subgroup is transitive. Every primitive permutation group is quasiprimitive, but the converse is not true. In this paper we start a project whose goal is to check which of the classical results on finite primitive permutation groups also holds for quasiprimitive ones (possibly with some modifications). The main topics addressed here are bounds on order, minimum degree and base size, as well as groups containing special p-elements. We also pose some problems for further research.

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