On Permutation Groups with Regular Subgroup

1974 ◽  
Vol 17 (3) ◽  
pp. 359-361
Author(s):  
R. D. Bercov
Keyword(s):  
Cyclic Group ◽  
Primitive Group ◽  
Permutation Groups ◽  
Regular Subgroup

W. Burnside [3, p. 343] showed that a cyclic group of order pm (p prime, m > l) cannot occur as a regular subgroup of a simply transitive primitive group. (For definitions and notation see [9].) Groups which are contained regularly in a primitive group G only when G is doubly transitive are therefore called B-groups [9, p. 64].

scholarly journals The Double Transitivity of a Class of Permutation Groups

1965 ◽  
Vol 17 ◽  
pp. 480-493 ◽  
Author(s):  
Ronald D. Bercov
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Transitive Group ◽  
Primitive Group ◽  
Permutation Groups ◽  
Regular Subgroup

Certain finite groups H do not occur as a regular subgroup of a uniprimitive (primitive but not doubly transitive) group G. If such a group H occurs as a regular subgroup of a primitive group G, it follows that G is doubly transitive. Such groups H are called B-groups (8) since the first example was given by Burnside (1, p. 343), who showed that a cyclic p-group of order greater than p has this property (and is therefore a B-group in our terminology).Burnside conjectured that all abelian groups are B-groups. A class of counterexamples to this conjecture due to W. A. Manning was given by Dorothy Manning in 1936 (3).

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.

scholarly journals Primitive permutation groups with a regular subgroup

2007 ◽  
Vol 310 (2) ◽  
pp. 569-618 ◽  
Author(s):  
Barbara Baumeister
Keyword(s):  
Permutation Groups ◽  
Regular Subgroup

scholarly journals Coset closure of a circulant S-ring and schurity problem

2016 ◽  
Vol 15 (04) ◽  
pp. 1650068 ◽  
Author(s):  
Sergei Evdokimov ◽  
Ilya Ponomarenko
Keyword(s):  
Finite Group ◽  
Linear System ◽  
Characteristic Property ◽  
Permutation Groups ◽  
Galois Correspondence ◽  
Regular Subgroup ◽  
A Finite Group

Let [Formula: see text] be a finite group. There is a natural Galois correspondence between the permutation groups containing [Formula: see text] as a regular subgroup, and the Schur rings (S-rings) over [Formula: see text]. The problem we deal with in the paper, is to characterize those S-rings that are closed under this correspondence, when the group [Formula: see text] is cyclic (the schurity problem for circulant S-rings). It is proved that up to a natural reduction, the characteristic property of such an S-ring is to be a certain algebraic fusion of its coset closure introduced and studied in the paper. Based on this characterization we show that the schurity problem is equivalent to the consistency of a modular linear system associated with a circulant S-ring under consideration. As a byproduct we show that a circulant S-ring is Galois closed if and only if so is its dual.

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].

Compact cellular algebras and permutation groups

1999 ◽  
Vol 197-198 (1-3) ◽  
pp. 247-267 ◽  
Author(s):  
S Evdokimov
Keyword(s):  
Permutation Groups ◽  
Cellular Algebras
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