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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 ) .

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.

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.

Abelian groups of zero adjoint entropy

L. Salce, P. Zanardo (2010)

Colloquium Mathematicae

The notion of adjoint entropy for endomorphisms of an Abelian group is somehow dual to that of algebraic entropy. The Abelian groups of zero adjoint entropy, i.e. ones whose endomorphisms all have zero adjoint entropy, are investigated. Torsion groups and cotorsion groups satisfying this condition are characterized. It is shown that many classes of torsionfree groups contain groups of either zero or infinite adjoint entropy. In particular, no characterization of torsionfree groups of zero adjoint...

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