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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 $G ≅ Sym (p + 1)$ .

A version of Brauer’s theorem for integer central functions

Fedor Bogomolov, Jorge Maciel (2009)

Open Mathematics

In this article we prove an effective version of the classical Brauer’s Theorem for integer class functions on finite groups.

A very short proof of Forester's rigidity result.

Guirardel, Vincent (2003)

Geometry & Topology

A Very Simple Proof of Bott's Theorem.

Michel Demazure (1976)

Inventiones mathematicae

A volume form on the SU(2)-representation space of knot groups.

Dubois, Jérôme (2006)

Algebraic & Geometric Topology

A weak type (1,1) estimate for a maximal operator on a group of isometries of a homogeneous tree

Michael G. Cowling, Stefano Meda, Alberto G. Setti (2010)

Colloquium Mathematicae

We give a simple proof of a result of R. Rochberg and M. H. Taibleson that various maximal operators on a homogeneous tree, including the Hardy-Littlewood and spherical maximal operators, are of weak type (1,1). This result extends to corresponding maximal operators on a transitive group of isometries of the tree, and in particular for (nonabelian finitely generated) free groups.

A $Z J$ -theorem for $p^{*}, p$ -injectors in finite groups

M. J. Iranzo, A. Martínez-Pastor, F. Pérez-Monasor (1992)

Rendiconti del Seminario Matematico della Università di Padova

A $λ$ -ring Frobenius characteristic for $G ≀ S_{n}$ .

Mendes, Anthony, Remmel, Jeffrey, Wagner, Jennifer (2004)

The Electronic Journal of Combinatorics [electronic only]

Abelian Extensions of Arbitrary Fields.

W. Kuyk, H.W. jr. Lenstra (1975)

Mathematische Annalen

Abelian Factorizations of Infinite Groups.

Bernhard Amberg (1971)

Mathematische Zeitschrift

Abelian group extensions and the axiom of constructibility.

Martin Huber, Paul C. Eklof (1979)

Commentarii mathematici Helvetici

Abelian group pairs having a trivial coGalois group

Paul Hill (2008)

Czechoslovak Mathematical Journal

Torsion-free covers are considered for objects in the category $q_{2} .$ Objects in the category $q_{2}$ are just maps in $R$ -Mod. For $R = ℤ,$ we find necessary and sufficient conditions for the coGalois group $G (A ⟶ B),$ associated to a torsion-free cover, to be trivial for an object $A ⟶ B$ in $q_{2} .$ Our results generalize those of E. Enochs and J. Rado for abelian groups.

Abelian groups have/are near Frattini subgroups

Simion Breaz, Grigore Călugăreanu (2002)

Commentationes Mathematicae Universitatis Carolinae

The notions of nearly-maximal and near Frattini subgroups considered by J.B. Riles in [20] and the natural related notions are characterized for abelian groups.

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A uniform chain whose inverse semigroup has no chart.

A uniformly quasiconformal group not isomorphic to a Möbius group.

A unipotent group associated with certain linear groups

A useful category for mixed Abelian groups.

A variation of Thompson's conjecture for the symmetric groups