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For a finite group $G$ denote by $N (G)$ the set of conjugacy class sizes of $G$ . In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N (G) = N (L)$ , then $G ≅ L$ . We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z (G) = 1$ and $N (G) = N (A_{i})$ is necessarily isomorphic to $A_{i}$ , where $i \in {2 p, 2 p + 1}$ .

Thompson's Problem (... =3).

Enrico Bombieri (1980)

Inventiones mathematicae

Three amalgams of A_5

Panagiotis Papadopoulos (1999)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

Three remarks on absolutely solvable groups.

Pálfy, Péter P. (2009)

Beiträge zur Algebra und Geometrie

Tits Geometries and Parabolic Systems in Finitely Generated Groups I.

Franz G. Timmesfeld (1983)

Mathematische Zeitschrift

Tits Geometries and Parabolic Systems in Finitely Generated Groups II.

Franz G. Timmesfeld (1983)

Mathematische Zeitschrift

TI-подгруппы в группах типа характеристики 2

А.А. Махнев (1985)

Matematiceskij sbornik

Topologische Automorphismen in der konstruktiven Galoistheorie.

Heinrich, B. Matzat (1986)

Journal für die reine und angewandte Mathematik

Torsion units for some almost simple groups

Joe Gildea (2016)

Czechoslovak Mathematical Journal

We investigate the Zassenhaus conjecture regarding rational conjugacy of torsion units in integral group rings for certain automorphism groups of simple groups. Recently, many new restrictions on partial augmentations for torsion units of integral group rings have improved the effectiveness of the Luther-Passi method for verifying the Zassenhaus conjecture for certain groups. We prove that the Zassenhaus conjecture is true for the automorphism group of the simple group $PSL (2, 11)$ . Additionally we prove that...

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The superfocal subgroup

The symmetric group of degree six can be covered by 13 and no fewer proper subgroups.

The Tits Group as a Standard Subgroup.

The uniqueness of groups of type J4.

The weights of irreducible ${SL}_{3} (q)$ -modules in the defining characteristic.

Théorèmes de Sylow génériques pour les groupes réductifs sur les corps finis.

Théorèmes sur les groupes de substitutions

Theta pairs and the structure of finite groups.

Thompson’s conjecture for the alternating group of degree $2 p$ and $2 p + 1$