A Note on Frobenius Groups

Flavell, Paul

doi:10.1006/jabr.2000.8269

Cited by 3 publications

(5 citation statements)

References 9 publications

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“…By the same reasoning, 5 U {1} is a right transversal for A 9 , for each g € G. It follows that every u, v € 5 U {1} satisfy uv~l G 5 U {1}. Hence S U {1} is a subgroup of G. 0 We do not know whether there exist primitive non-freely transitive groups.…”

Section: If the Centre Of K Is Trivial Then We Can Consider A" As A Smentioning

confidence: 99%

“…Let G be an abstract group with a subgroup A. We shall say that A is closed in G if A is a maximal subgroup in the set \J A 9 . That is, if for any subgroup B such that B > A it holds that sec B % |J A 9 .…”

Section: F R E E L Y Transitive Groups and Their Basic Propertiesmentioning

confidence: 99%

“…Then [G : A] = n. We know that A n A 9 = 1 for every g € G -A. An elementary computation shows that the set of fixed-point-free elements, that is, the set S -G -\J A 9 , has order n -1.…”

Section: If the Centre Of K Is Trivial Then We Can Consider A" As A Smentioning

confidence: 99%

“…Assume then that \G\ is even and let u € G be an involution. We may assume that u is not in the centre of G. Since G is global, there exists a set T which is a right transversal for all the conjugates (u 9 ) in G. Notice that for any g e G, u 9 T = G -T, whence for any g,h G G we have u 9 T = u h T and u 9 u h Ker(T) = {k e G \ kT = T}. This is a proper subgroup of G (whose order divides \T\), and it contains the set 5 = {u 9 u h \ g, h € G}.…”

Section: Degreementioning

confidence: 99%

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Transversals in permutation groups and factorisations of complete graphs

Kaplan

Lev

2002

Bull. Austral. Math. Soc.

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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.

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Section: If the Centre Of K Is Trivial Then We Can Consider A" As A Smentioning

confidence: 99%

Section: F R E E L Y Transitive Groups and Their Basic Propertiesmentioning

confidence: 99%

“…Then [G : A] = n. We know that A n A 9 = 1 for every g € G -A. An elementary computation shows that the set of fixed-point-free elements, that is, the set S -G -\J A 9 , has order n -1.…”

Section: If the Centre Of K Is Trivial Then We Can Consider A" As A Smentioning

confidence: 99%

Section: Degreementioning

confidence: 99%

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Transversals in permutation groups and factorisations of complete graphs

Kaplan

Lev

2002

Bull. Austral. Math. Soc.

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“…If G is a finite group and H, K are malnormal subgroups of G, then there exists an element g such that H g K or K g H (P. Flavell [3]). The situation with malnormal subgroups in infinite groups much more complica ted.…”

mentioning

confidence: 99%

A generalization of the malnormal subgroups

Kurdachenko¹,

Semko²,

Subbotin³

2019

Dopov. Nac. akad. nauk Ukr.

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A generalization of the malnormal subgroups Presented by Academician of the NAS of Ukraine A.M. Samoilenko A subgroup H of a group G is called malonormal in G, if H ∩ H x = 〈1〉 for every element x ∉ N G (H). These subgroups are generalizations of malnormal subgroups. Every malnormal subgroup is malonormal, and every self-normalizing malonormal subgroup is malnormal. Furthermore, every normal subgroup is malonormal. In this paper we obtain a description of finite and certain infinite groups, whose subgroups are malonormal.

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A note on Flavell’s theorem associated with Frobenius groups

He,

Zhu

2023

Communications in Algebra

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A Note on Frobenius Groups

Cited by 3 publications

References 9 publications

Transversals in permutation groups and factorisations of complete graphs

Transversals in permutation groups and factorisations of complete graphs

A generalization of the malnormal subgroups

A note on Flavell’s theorem associated with Frobenius groups

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