On transitive permutation groups with a subgroup satisfying a certain conjugacy condition

1984 ◽  
Vol 36 (1) ◽  
pp. 69-86 ◽  
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Permutation Group ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Pronormal Subgroup ◽  
Natural Representation ◽  
Conjugacy Condition ◽  
Sylow Subgroups ◽  
Point Stabilizer

AbstractLet G be transitive permutation group of degree n and let K be a nontrivial pronormal subgroup of G (that is, for all g in G, K and Kg are conjugate in (K, Kg)). It is shown that K can fix at most ½(n – 1) points. Moreover if K fixes exactly ½(n – 1) points then G is either An or Sn, or GL(d, 2) in its natural representation where n = 2d-1 ≥ 7. Connections with a result of Michael O'Nan are dicussed, and an application to the Sylow subgroups of a one point stabilizer is given.

On the fixed points of Sylow subgroups of transitive permutation groups

1976 ◽  
Vol 21 (4) ◽  
pp. 428-437 ◽  
Author(s):  
Marcel Herzog ◽  
Cheryl E. Praeger
Keyword(s):  
Fixed Points ◽  
Permutation Group ◽  
Upper Bound ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Sylow Subgroups

AbstractLet G be a transitive permutation group on a set Ω of n points, and let P be a Sylow p-subgroup of G for some prime p dividing ∣G∣. If P has t long orbits and f fixed points in Ω, then it is shown that f ≦ tp − ip(n), where ip(n) = p – rp(n), rp(n) denoting the residue of n modulo p. In addition, groups for which f attains the upper bound are classified.

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.

Some generalized characters associated to a transitive permutation group

2020 ◽  
Vol 23 (3) ◽  
pp. 393-397
Author(s):  
Wolfgang Knapp ◽  
Peter Schmid
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Frobenius Group ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Transitive Permutation Group ◽  
Point Stabilizer ◽  
Necessary And Sufficient ◽  
Regular Character ◽  
Permutation Character

AbstractLet G be a finite transitive permutation group of degree n, with point stabilizer {H\neq 1} and permutation character π. For every positive integer t, we consider the generalized character {\psi_{t}=\rho_{G}-t(\pi-1_{G})}, where {\rho_{G}} is the regular character of G and {1_{G}} the 1-character. We give necessary and sufficient conditions on t (and G) which guarantee that {\psi_{t}} is a character of G. A necessary condition is that {t\leq\min\{n-1,\lvert H\rvert\}}, and it turns out that {\psi_{t}} is a character of G for {t=n-1} resp. {t=\lvert H\rvert} precisely when G is 2-transitive resp. a Frobenius group.

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.

scholarly journals On doubly transitive permutation groups of degree prime squared plus one

1977 ◽  
Vol 23 (2) ◽  
pp. 202-206 ◽  
Author(s):  
David Chillag
Keyword(s):  
Permutation Group ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Normal Extension ◽  
Stabilizer Of A Point

AbstractA doubly transitive permutation group of degreep2+ 1, pa prime, is proved to be doubly primitive forp≠ 2. We also show that if such a group is not triply transitive then either it is a normal extension ofP S L(2,p2) or the stabilizer of a point is a rank 3 group.

scholarly journals On the normal structure of a one-point stabilizer in a doubly transitive permutation group

1978 ◽  
Vol 25 (2) ◽  
pp. 145-166
Author(s):  
M. D. Atkinson ◽  
Cheryl E. Praeger
Keyword(s):  
Normal Subgroup ◽  
Permutation Group ◽  
Closed Subgroup ◽  
Normal Structure ◽  
Subject Classification ◽  
Transitive Permutation Group ◽  
Normal Extension ◽  
Extra Information ◽  
Finite Set ◽  
Point Stabilizer

Let G be a doubly transitive permutation group on a finite set Ω, and let Kα be a normal subgroup of the stabilizer Gα of a point α in Ω. If the action of Gα on the set of orbits of Kα in Ω − {α} is 2-primitive with kernel Kα it is shown that either G is a normal extension of PSL(3, q) or Kα ∩ Gγ is a strongly closed subgroup of Gαγ in Gα, where γ ∈ Ω − {α}. If in addition the action of Gα on the set of orbits of Kα is assumed to be 3-transitive, extra information is obtained using permutation theoretic and centralizer ring methods. In the case where Kα has three orbits in Ω − {α} strong restrictions are obtained on either the structure of G or the degrees of certain irreducible characters of G. Subject classification (Amer. Math. Soc. (MOS) 1970: 20 B 20, 20 B 25.

scholarly journals Transitive Extensions of a Class of Doubly Transitive Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 159-169 ◽  
Author(s):  
Michio Suzuki
Keyword(s):  
Recent Result ◽  
Permutation Group ◽  
Unitary Group ◽  
Simple Groups ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Precise Statement ◽  
Transitive Groups ◽  
Recent Theorem ◽  
Doubly Transitive Groups

1. When a permutation group G on a set Ω is given, a transitive extension G of G is defined to be a transitive permutation group on the set Γ which is a union of Ω and a new point ∞ such that the stabilizer of ∞ in G1 is isomorphic to G as a permutation group on Ω. The purpose of this paper is to prove that many known simple groups which can be represented as doubly transitive groups admit no transitive extension. Precise statement is found in Theorem 2. For example, the simple groups discovered by Ree [5] do not admit transitive extensions. Theorem 2 includes also a recent result of D. R. Hughes [3] which states that the unitary group U3(q) q>2 does not admit a transitive extension. As an application we prove a recent theorem of H. Nagao [4], which generalizes a theorem of Wielandt on the non-existence of 8-transitive permutation groups not containing the alternating groups under Schreier’s conjecture.

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