Wreath product in automorphism groups of graphs

Grech, Mariusz; Kisielewicz, Andrzej

doi:10.1002/jgt.22808

Cited by 1 publication

(2 citation statements)

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“…On the other hand, in [7,Corollary 5.3], they stated that there are infinitely many examples of imprimitive permutation groups that are orbit closed but not a relation group. Unfortunately, as it was noted in [13], the proof of this claim contains an error. Given as an example the group C 3 L and other examples suggested in the proof are indeed not relation groups, but they are not orbit closed either.…”

Section: Introductionmentioning

confidence: 94%

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Orbit closed permutation groups, relation groups, and simple groups

Grech

Kisielewicz

2023

J Algebr Comb

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A permutation group G on a set $$\Omega $$ Ω is called orbit closed if every permutation of $$\Omega $$ Ω preserving the orbits of G in its action on the power set $$P(\Omega )$$ P ( Ω ) belongs to G. It is called a relation group if there exists a family $$R \subseteq P(\Omega )$$ R ⊆ P ( Ω ) such that G is the group of all permutations preserving R. We prove that if a finite orbit closed permutation group G is simple, or is a subgroup of a simple group, then it is a relation group. This result justifies our general conjecture that with a few exceptions every finite orbit closed group is a relation group. To obtain the result, we prove that most of the finite simple permutation groups are relation groups. We also obtain a complete description of those finite simple permutation groups that have regular sets, and prove that (with one exception) if a finite simple permutation group G is a relation group, then every subgroup of G is a relation group.

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Section: Introductionmentioning

confidence: 94%

“…5,11)(6,7,9)(8,12,13) ψ = (3,8, 7)(5, 12, 9)(6, 11, 13) (1, 13, 7)(2, 10, 6)(3, 5, 12)(4, 11, 9) ψ = (1, 13, 7)(2, 10, 6)(3, 5, 12)(4, 11, 9) c) L 4 (2) (regular sets of sizes from 6 to 24. ):…”

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confidence: 99%