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Let $G$ be a finite group and $nse (G)$ the set of numbers of elements with the same order in $G$ . In this paper, we prove that a finite group $G$ is isomorphic to $M$ , where $M$ is one of the Mathieu groups, if and only if the following hold: (1) $| G | = | M |$ , (2) $nse (G) = nse (M)$ .

A Root System for the Lyons Group.

Werner Meyer, Wolfram Neutsch (1989)

Mathematische Annalen

A simple construction for the Fischer-Griess monster group.

J.H. Conway (1985)

Inventiones mathematicae

A variation of Thompson's conjecture for the symmetric groups

Mahdi Abedei, Ali Iranmanesh, Farrokh Shirjian (2020)

Czechoslovak Mathematical Journal

Let $G$ be a finite group and let $N (G)$ denote the set of conjugacy class sizes of $G$ . Thompson’s conjecture states that if $G$ is a centerless group and $S$ is a non-abelian simple group satisfying $N (G) = N (S)$ , then $G ≅ S$ . In this paper, we investigate a variation of this conjecture for some symmetric groups under a weaker assumption. In particular, it is shown that $G ≅ Sym (p + 1)$ if and only if $| G | = (p + 1)!$ and $G$ has a special conjugacy class of size $(p + 1)! / p$ , where $p > 5$ is a prime number. Consequently, if $G$ is a centerless group with $N (G) = N (Sym (p + 1))$ , then $G ≅ Sym (p + 1)$ .

All geometries of the Mathieu group $M_{11}$ based on maximal subgroups.

Buekenhout, Francis, Dehon, Michel, Leemans, Dimitri (1996)

Experimental Mathematics

An Amalgam Related to M_11 and SP_4(3)

Panagiotis Papadopoulos (2002)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

An existence lemma for groups generated by 3-transpositions.

Richard Weiss, John van Bon (1992)

Inventiones mathematicae

Characterization of the alternating groups by their order and one conjugacy class length

Alireza Khalili Asboei, Reza Mohammadyari (2016)

Czechoslovak Mathematical Journal

Let $G$ be a finite group, and let $N (G)$ be the set of conjugacy class sizes of $G$ . By Thompson’s conjecture, if $L$ is a finite non-abelian simple group, $G$ is a finite group with a trivial center, and $N (G) = N (L)$ , then $L$ and $G$ are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation). In...

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2-signalizers of finite simple groups.

3-локальная характеризация группы Хельда

3-характеризация группы О'Нэна - Симса

A block-theory-free characterization of $M_{24}$

A characterization of the groups Fi22, Fi23 and F24.

A characterization theorem for sporadic simple groups.

A Geometric Construction of Janko's Group J... .

A group-free characterization of the $P$ -geometry for ${Co}_{2}$ .

A new characterization of Mathieu groups

A Root System for the Lyons Group.

A simple construction for the Fischer-Griess monster group.

A variation of Thompson's conjecture for the symmetric groups

All geometries of the Mathieu group $M_{11}$ based on maximal subgroups.

An Amalgam Related to M_11 and SP_4(3)

An existence lemma for groups generated by 3-transpositions.

Characterization of the alternating groups by their order and one conjugacy class length

Characterizing the Sporadic Simple Group of Suzuki by a 2-Local Subgroup.

Conway's Group CO3 and the Dickson Invariants.

Coset enumeration of groups generated by symmetric sets of involutions.

Double-coset enumeration algorithm for symmetrically generated groups.

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