scholarly journals On Primitive Extensions of Rank 3 of Symmetric Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 171-177 ◽  
Author(s):  
Tosiro Tsuzuku
Keyword(s):  
Permutation Group ◽  
Symmetric Groups ◽  
Transitive Group ◽  
Permutation Groups ◽  
Minimal Set ◽  
Finite Set ◽  
Rank 2 ◽  
One To One

1. Let Ω be a finite set of arbitrary elements and let (G, Ω) be a permutation group on Ω. (This is also simply denoted by G). Two permutation groups (G, Ω) and (G, Γ) are called isomorphic if there exist an isomorphism σ of G onto H and a one to one mapping τ of Ω onto Γ such that (g(i))τ=gσ(iτ) for g ∊ G and i∊Ω. For a subset Δ of Ω, those elements of G which leave each point of Δ individually fixed form a subgroup GΔ of G which is called a stabilizer of Δ. A subset Γ of Ω is called an orbit of GΔ if Γ is a minimal set on which each element of G induces a permutation. A permutation group (G, Ω) is called a group of rank n if G is transitive on Ω and the number of orbits of a stabilizer Ga of a ∊ Ω, is n. A group of rank 2 is nothing but a doubly transitive group and there exist a few results on structure of groups of rank 3 (cf. H. Wielandt [6], D. G. Higman M).

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 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 Wreath decompositions of finite permutation groups

1989 ◽  
Vol 40 (2) ◽  
pp. 255-279 ◽  
Author(s):  
L. G. Kovács
Keyword(s):  
Permutation Group ◽  
Wreath Product ◽  
Transitive Group ◽  
Wreath Products ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Formal Definition ◽  
The Third ◽  
Product Action ◽  
Factor Decomposition

There is a familiar construction with two finite, transitive permutation groups as input and a finite, transitive permutation group, called their wreath product, as output. The corresponding ‘imprimitive wreath decomposition’ concept is the first subject of this paper. A formal definition is adopted and an overview obtained for all such decompositions of any given finite, transitive group. The result may be heuristically expressed as follows, exploiting the associative nature of the construction. Each finite transitive permutation group may be written, essentially uniquely, as the wreath product of a sequence of wreath-indecomposable groups, amid the two-factor wreath decompositions of the group are precisely those which one obtains by bracketing this many-factor decomposition.If both input groups are nontrivial, the output above is always imprimitive. A similar construction gives a primitive output, called the wreath product in product action, provided the first input group is primitive and not regular. The second subject of the paper is the ‘product action wreath decomposition’ concept dual to this. An analogue of the result stated above is established for primitive groups with nonabelian socle.Given a primitive subgroup G with non-regular socle in some symmetric group S, how many subgroups W of S which contain G and have the same socle, are wreath products in product action? The third part of the paper outlines an algorithm which reduces this count to questions about permutation groups whose degrees are very much smaller than that of G.

S-rings over loops, right mapping groups and transversals in permutation groups

1981 ◽  
Vol 89 (3) ◽  
pp. 433-443 ◽  
Author(s):  
K. W. Johnson
Keyword(s):  
Permutation Group ◽  
Group Ring ◽  
Permutation Representation ◽  
Permutation Groups ◽  
P 75 ◽  
Regular Subgroup ◽  
Finite Set

The centralizer ring of a permutation representation of a group appears in several contexts. In (19) and (20) Schur considered the situation where a permutation group G acting on a finite set Ω has a regular subgroup H. In this case Ω may be given the structure of H and the centralizer ring is isomorphic to a subring of the group ring of H. Schur used this in his investigations of B-groups. A group H is a B-group if whenever a permutation group G contains H as a regular subgroup then G is either imprimitive or doubly transitive. Surveys of the results known on B-groups are given in (28), ch. IV and (21), ch. 13. In (28), p. 75, remark F, it is noted that the existence of a regular subgroup is not necessary for many of the arguments. This paper may be regarded as an extension of this remark, but the approach here differs slightly from that suggested by Wielandt in that it appears to be more natural to work with transversals rather than cosets.

Trivial Set-Stabilizers in Finite Permutation Groups

1983 ◽  
Vol 35 (1) ◽  
pp. 59-67 ◽  
Author(s):  
David Gluck
Keyword(s):  
Permutation Group ◽  
Symmetric Groups ◽  
Infinite Family ◽  
Permutation Groups ◽  
Alternating Groups ◽  
Sylow Subgroups ◽  
Odd Order

For which permutation groups does there exist a subset of the permuted set whose stabilizer in the group is trivial?The permuted set has so many subsets that one might expect that subsets with trivial stabilizer usually exist. The symmetric and alternating groups are obvious exceptions to this expectation. Another, more interesting, infinite family of exceptions are the 2-Sylow subgroups of the symmetric groups on 2n symbols, in their natural representations on 2n points.One of our main results, Corollary 1, sheds some light on this last family of groups. We show that when the permutation group has odd order, there is indeed a subset of the permuted set whose stabilizer in the group is trivial. Corollary 1 follows easily from Theorem 1, which completely classifies the primitive solvable permutation groups in which every subset of the permuted set has non-trivial stabilizer.

scholarly journals Solvability of a Class of Rank 3 Permutation Groups

1971 ◽  
Vol 41 ◽  
pp. 89-96 ◽  
Author(s):  
D.G. Higman
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Regular Graph ◽  
Strongly Regular Graph ◽  
Permutation Groups ◽  
Theoretical Interpretation ◽  
Finite Set ◽  
Strongly Regular ◽  
Even Order

1. Introduction. Let G be a rank 3 permutation group of even order on a finite set X, |X| = n, and let Δ and Γ be the two nontrivial orbits of G in X×X under componentwise action. As pointed out by Sims [6], results in [2] can be interpreted as implying that the graph = (X, Δ) is a strongly regular graph, the graph theoretical interpretation of the parameters k, l, λ and μ of [2] being as follows: k is the degree of , λ is the number of triangles containing a given edge, and μ is the number of paths of length 2 joining a given vertex P to each of the l vertices ≠ P which are not adjacent to P. The group G acts as an automorphism group on and on its complement = (X,Γ).

scholarly journals Sylow subgroups of transitive permutation groups II

1977 ◽  
Vol 23 (3) ◽  
pp. 329-332 ◽  
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Permutation Group ◽  
Minimal Length ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Sylow Subgroups ◽  
Finite Set

AbstractLet G be a transitive permutation group on a finite set of n points, and let P be a Sylow p-subgroup of G for some prime p dividing |G|. We are concerned with finding a bound for the number f of points of the set fixed by P. Of all the orbits of P of length greater than one, suppose that the ones of minimal length have length q, and suppose that there are k orbits of P of length q. We show that f ≦ kp − ip(n), where ip(n) is the integer satisfying 1 ≦ ip(n) ≦ p and n + ip(n) ≡ 0(mod p). This is a generalisation of a bound found by Marcel Herzog and the author, and this new bound is better whenever P has an orbit of length greater than the minimal length q.

2-closures of primitive permutation groups of holomorph type

2019 ◽  
Vol 17 (1) ◽  
pp. 795-801 ◽  
Author(s):  
Xue Yu ◽  
Jiangmin Pan
Keyword(s):  
Permutation Group ◽  
Permutation Groups ◽  
Finite Set

Abstract The 2-closure G(2) of a permutation group G on a finite set Ω is the largest subgroup of Sym(Ω) which has the same orbits as G in the induced action on Ω × Ω. In this paper, the 2-closures of certain primitive permutation groups of holomorph simple and holomorph compound types are determined.

scholarly journals On doubly transitive permutation groups

1978 ◽  
Vol 18 (3) ◽  
pp. 465-473 ◽  
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Permutation Group ◽  
Collineation Group ◽  
Affine Plane ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Nonidentity Element ◽  
Finite Set

Suppose that G is a doubly transitive permutation group on a finite set Ω and that for α in ω the stabilizer Gα of αhas a set σ = {B1, …, Bt} of nontrivial blocks of imprimitivity in Ω – {α}. If Gα is 3-transitive on σ it is shown that either G is a collineation group of a desarguesian projective or affine plane or no nonidentity element of Gα fixes B pointwise.

scholarly journals Asymptotically free action of permutation groups on subsets and multisets

2015 ◽  
Vol 25 (1) ◽  
Author(s):  
Sergey Yu. Sadov
Keyword(s):  
Permutation Group ◽  
Free Action ◽  
Permutation Groups ◽  
Finite Set

AbstractLet G be a permutation group acting on a finite set Ω of cardinality n. The number of orbits of the induced action of G on the set Ω

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