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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 $G ≅ Sym (p + 1)$ .

Displaying 1 – 12 of 12

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.

Currently displaying 1 – 12 of 12