§ 295 Any irregular p-group contains a non-isolated maximal regular subgroup

2018 ◽  
pp. 176-177
Keyword(s):  
Regular Subgroup

scholarly journals Permutation groups containing a regular abelian subgroup: the tangled history of two mistakes of Burnside

2019 ◽  
Vol 168 (3) ◽  
pp. 613-633 ◽  
Author(s):  
MARK WILDON
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Open Problems ◽  
Related Literature ◽  
Cyclic Groups ◽  
Computational Data ◽  
Regular Subgroup ◽  
History Of ◽  
Order State

AbstractA group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later, in 1921, he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this paper we explain these flaws and prove that every cyclic group of composite order is a B-group, using only Burnside’s character-theoretic methods. We also survey the related literature, prove some new results on B-groups of prime-power order, state two related open problems and present some new computational data.

Archimedean Closures in Lattice-Ordered Groups

1969 ◽  
Vol 21 ◽  
pp. 1004-1012 ◽  
Author(s):  
Richard D. Byrd
Keyword(s):  
Partial Answer ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Regular Subgroup

Conrad (10) and Wolfenstein (15; 16) have introduced the notion of an archimedean extension (a-extension) of a lattice-ordered group (l-group). In this note the class of l-groups that possess a plenary subset of regular subgroups which are normal in the convex l-subgroups that cover them are studied. It is shown in § 3 (Corollary 3.4) that the class is closed with respect to a-extensions and (Corollary 3.7) that each member of the class has an a-closure. This extends (6, p. 324, Corollary II; 10, Theorems 3.2 and 4.2; 15, Theorem 1) and gives a partial answer to (10, p. 159, Question 1). The key to proving both of these results is Theorem 3.3, which asserts that if a regular subgroup is normal in the convex l-subgroup that covers it, then this property is preserved by a-extensions.

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

C4C8(S) tori which are Cayley graphs

2020 ◽  
Vol 12 (06) ◽  
pp. 2050073
Author(s):  
Chunqi Liu
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Regular Subgroup

A [Formula: see text] net is a trivalent decoration made by alternating square [Formula: see text] and octagons [Formula: see text]. It can cover either a cylinder or a tori. Cayley graph [Formula: see text] on a group [Formula: see text] with connection set [Formula: see text] has the elements of [Formula: see text] as its vertices and an edge joining [Formula: see text] and [Formula: see text] for all [Formula: see text] and [Formula: see text]. Motivated by Afshari’s work, we show that the [Formula: see text] tori are Cayley graphs by constructing a regular subgroup of the automorphism group of [Formula: see text].

An investigation of countable B-groups

1987 ◽  
Vol 102 (2) ◽  
pp. 223-231 ◽  
Author(s):  
P. Cameron ◽  
K. W. Johnson
Keyword(s):  
Permutation Group ◽  
Countable Group ◽  
General Question ◽  
Primitive Permutation Group ◽  
Square Root ◽  
Regular Subgroup

A group G is defined to be a B-group if any primitive permutation group which contains G as a regular subgroup is doubly transitive. In the case where G is finite the existence of families of B-groups has been established by Burnside, Schur, Wielandt and others and led to the investigation of S-rings. A survey of this work is given in [3], sections 13·7–13·12. In this paper the possibility of the existence of countable B-groups is discussed. Three distinct methods are given to embed a countable group as a regular subgroup of a simply primitive permutation group, and in each case a condition on the square root sets of elements of the group is necessary for the embedding to be carried out. It is easy to demonstrate that this condition is not sufficient, and the general question remains open.

scholarly journals Fixed Point Free Actions of Groups of Exponent 5

2004 ◽  
Vol 77 (3) ◽  
pp. 297-304 ◽  
Author(s):  
Enrico Jabara
Keyword(s):  
Fixed Point ◽  
Vector Space ◽  
Regular Subgroup ◽  
Free Actions

AbstractIn this paper we prove that if V is a vector space over a field of positive characteristric p ≠ 5 then any regular subgroup A of exponent 5 of GL(V) is cyclic. As a consequence a conjecture of Gupta and Mazurov is proved to be true.

scholarly journals Cubic symmetric graphs of order twice an odd prime-power

2006 ◽  
Vol 81 (2) ◽  
pp. 153-164 ◽  
Author(s):  
Yan-Quan Feng ◽  
Jin Ho Kwak
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Prime Power ◽  
Symmetric Graph ◽  
Cubic Graphs ◽  
Full Automorphism Group ◽  
Symmetric Graphs ◽  
Regular Subgroup

AbstractAn automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p, we show that if p ≠ 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p ≠3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s − 1)-regular subgroup for each 1 ≦ s ≦ 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1≦ s≦ 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.

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