Distinguishing simple groups

Grech, Mariusz; Kisielewicz, Andrzej

doi:10.48550/arxiv.2001.06300

Cited by 2 publications

(9 citation statements)

References 21 publications

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“…There are many various actions of simple groups, transitive and intransitive, and our result [8] describes all actions with the distinguished number larger than 2. Since the kernel of the action of a group G is a normal subgroup, it follows that all actions of simple groups are faithful.…”

Section: Introductionmentioning

confidence: 78%

“…If G = H and the isomorphism ψ the identity, we write G (2) for G|| ψ G, and more generally, G (k) in the case of k ≥ 1 summands. We call it the parallel multiple of the permutation group G and adopt the convention G (1) = G. (We refer the reader to [8] for a more general construction).…”

Section: Introductionmentioning

confidence: 99%

“…Recall that a group G is simple if it has no nontrivial normal subgroup. There is a huge literature on finite simple groups connected with the classification of finite simple groups and some of these results have been used in our previous paper [8]. In this paper we apply only the main result of [8].…”

Section: Introductionmentioning

confidence: 99%

“…There is a huge literature on finite simple groups connected with the classification of finite simple groups and some of these results have been used in our previous paper [8]. In this paper we apply only the main result of [8]. It needs some preliminary explanation.…”

Section: Introductionmentioning

confidence: 99%

“…Also, various more involved constructions lead to such graphs. Yet, in general, we know very little about graphs whose automorphism groups are simple (see examples and an open problem on this topic in [8]). Our approach in this paper is to consider only those simple permutation groups that have the distinguishing number larger than 2 and examine them whether they may or may not be the automorphism groups of graphs.…”

Section: Introductionmentioning

confidence: 99%

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Asymmetric edge-coloring of graphs with simple automorphism group

Grech¹,

Kisielewicz²

2021

Preprint

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The distinguishing index D ′ (Γ) of a graph Γ is the least number k such that Γ has an edge-coloring with k colors preserved only by the trivial automorphism. In this paper we prove that if the automorphism group of a finite graph Γ is simple, then its distinguishing index D ′ (Γ) = 2.The distinguishing index of a graph Γ has been introduced by Pilśniak and Kalinowski [9] in 2015 to be the least number d such that Γ has an edge-coloring with d colors breaking the symmetry of Γ (i.e., such that no nontrivial automorphism of Γ preserves this coloring). This is an analog to the notion of the distinguishing number D(Γ) of a graph introduced by Albertson and Collins [1] in 1996, which has been defined in the same way for colorings of vertices.Note that for asymmetric graphs we have D(Γ) = D ′ (Γ) = 1. For other graphs D(Γ) ≥ 2, and it is conjectublack that almost all of them have the distinguishing number two (see [4,9]). The situation is similar for the distinguishing index, and the claim that having the distinguishing index two is generic for asymmetric graphs has been supported by results in [10,11,14].The concepts of the distinguished number and distinguished index generalize naturally to the distinguishing number of an arbitrary group action ([15, 3]). Following this generalization, it was realized in [2] that in permutation group theory the problem had been investigated for many years as a part of the study of set stabilisers of group actions, and some results may be successfully applied. In particular, a result by Gluck [6] (obtained as early as in 1983) shows that if the order of the automorphism group of a graph Γ is odd (and > 1), then both the distinguishing number and the distinguishing index of Γ are two,This paper is the sequel of [8], where we have proven, in particular, that if the automorphism group of a finite graph Γ is simple, then its distinguishing number D(Γ) = 2. To obtain this result we have described the distinguishing number for all possible actions of simple groups. Now, we apply the latter to show that for such graphs also the distinguishing index D ′ (Γ) = 2. This is not so straightforward as in the case of Gluck's result. It requires to consider

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Section: Introductionmentioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Asymmetric edge-coloring of graphs with simple automorphism group

Grech¹,

Kisielewicz²

2021

Preprint

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Distinguishing actions of symmetric groups and related graphs

Grech,

Kisielewicz

2020

Preprint

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The distinguishing number D(G, X) of an action of a group G on a set X is the least size of a partition of X such that no element of G acting nontrivially on X preserves this partition. In this paper we describe the distinguishing numbers for all actions of the symmetric group Sn, for any n ≥ 3. This allows us to describe the distinguishing numbers for all graphs whose automorphism group is isomorphic with a symmetric group. Our description solves a few open problems posed by various authors in earlier papers on this topic.

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Distinguishing simple groups

Cited by 2 publications

References 21 publications

Asymmetric edge-coloring of graphs with simple automorphism group

Asymmetric edge-coloring of graphs with simple automorphism group

Distinguishing actions of symmetric groups and related graphs

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