The admissibility of sporadic simple groups

Evans, Anthony B.

doi:10.1016/j.jalgebra.2008.09.028

Cited by 48 publications

(41 citation statements)

References 10 publications

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“…(3 · M 22 :2) (the third in the list in [12], and the second in the www-Atlas [15], from which information about the group G will be taken). Note that the existence of a complete mapping of H follows from the earlier results of [9,14]. Now H is the full centraliser in G ∼ = J 4 of a 2A-involution (J 4 -class), x say, so that the actions of G on the right cosets of H and on the conjugates of x are isomorphic, with Hg corresponding to x g .…”

Section: Preliminariesmentioning

confidence: 89%

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The Hall–Paige conjecture, and synchronization for affine and diagonal groups

et al. 2020

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The Hall-Paige conjecture asserts that a finite group has a complete mapping if and only if its Sylow subgroups are not cyclic. The conjecture is now proved, and one aim of this paper is to document the final step in the proof (for the sporadic simple group J 4 ).We apply this result to prove that primitive permutation groups of simple diagonal type with three or more simple factors in the socle are non-synchronizing. We also give the simpler proof that, for groups of affine type, or simple diagonal type with two socle factors, synchronization and separation are equivalent.Synchronization and separation are conditions on permutation groups which are stronger than primitivity but weaker than 2-homogeneity, the second of these being stronger than the first. Empirically it has been found that groups which are synchronizing but not separating are rather rare. It follows from our results that such groups must be primitive of almost simple type.

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Section: Preliminariesmentioning

confidence: 89%

“…Wilcox [14] reduced the conjecture to the case of simple groups, and proved it for groups of Lie type except for the Tits group. Evans [9] handled the Tits group and all the sporadic groups except J 4 . The proof in the final case was announced by the first author; we give details here.…”

Section: Preliminariesmentioning

confidence: 99%

The Hall–Paige conjecture, and synchronization for affine and diagonal groups

et al. 2020

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“…Given a group G, let Syl 2 (G) denote the isomorphism class of its Sylow 2-subgroups. The groups for which χ(G) = n were recently characterized by Bray, Evans, and Wilcox [5,16], resolving a 50 year old conjecture due to Hall and Paige [7]. Theorem 1.5.…”

Section: Introductionmentioning

confidence: 90%

“…To prove the claim, it is left to show that E(Γ i ) ∩ E S contains exactly n edges, each of which connects opposite vertices (i.e. vertices at distance q) in the cycle given by (5). Given an integer z, let z be the corresponding residue modulo t. It follows from (4) that, for every j ∈ [n],…”

Section: The Chromatic Number Of Abelian Groupsmentioning

confidence: 99%

The Chromatic Number of Finite Group Cayley Tables

Goddyn

Halasz

Mahmoodian

2019

Electron. J. Combin.

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The chromatic number of a latin square L, denoted χ(L), is the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies χ(L) ≤ |L|+2. If true, this would resolve a longstanding conjecture-commonly attributed to Brualdi-that every latin square has a partial transversal of size |L| − 1. Restricting our attention to Cayley tables of finite groups, we prove two main results. First, we resolve the chromatic number question for Cayley tables of finite Abelian groups: the Cayley table of an Abelian group G has chromatic number |G| or |G| + 2, with the latter case occurring if and only if G has nontrivial cyclic Sylow 2-subgroups. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups. For |G| ≥ 3, this improves the best-known general upper bound from 2|G| to 3 2 |G|, while yielding an even stronger result in infinitely many cases.We refer to a partition of a latin square L into k partial transversals as a (proper) k-coloring of L. The chromatic number of L, denoted χ(L), is the minimum k for which L has a k-coloring. As a partial transversal has size at most n, χ(L) ≥ n. On the other hand, we may bound χ(L) from above by applying Brooks' theorem to the graph Γ(L). Proposition 1.1. Let L be a latin square of order n ≥ 3. Thenwith equality holding for the lower bound if and only if L possesses an orthogonal mate.Parts of this work have already appeared in the M.Sc. thesis of the second author [6], which was supervised by the first author. For any undefined terms we refer the reader to [2,9,11]. Although latin square colorings are a natural generalization of the notion of possessing an orthogonal mate, the concept did not appear in the literature until very recently. In 2015, papers introducing the concept were submitted by two separate groups: Cavenagh and Kuhl [3] and Besharati et al.[1] (a group containing the first and third author of the present paper). The two groups independently conjectured the following upper bound.Conjecture 1.2. Let L be a latin square of order n. ThenLatin squares for which this conjecture is tight have been given by Euler [4] in the even case and by Wanless and Webb [15] in the odd case. If true, Conjecture 1.2 is likely difficult to prove, as it implies a pair of long-standing conjectures concerning the existence of large partial transversals in latin squares. These conjectures are attributed to Brualdi and Ryser, respectively. Conjecture 1.3 ([9, 12, 13]). Let L be a latin square of order n. Then (1) L possesses a near transversal, and (2) if n is odd then L possesses a transversal.

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“…Corollary 4. Any 2-transitive group has remoteness n. If G acts regularly, then the Hall-Paige conjecture [12] proved in [3,9,19] implies that r(G) = n− 1 if and only if its Sylow 2-subgroup is non-cyclic.…”

Section: Transitive Groups Proposition 12mentioning

confidence: 99%