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Self-small products of abelian groups

Josef Dvořák, Jan Žemlička (2022)

Commentationes Mathematicae Universitatis Carolinae

Let A and B be two abelian groups. The group A is called B -small if the covariant functor Hom ( A , - ) commutes with all direct sums B ( κ ) and A is self-small provided it is A -small. The paper characterizes self-small products applying developed closure properties of the classes of relatively small groups. As a consequence, self-small products of finitely generated abelian groups are described.

Simple proofs of some generalizations of the Wilson’s theorem

Jan Górowski, Adam Łomnicki (2014)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

In this paper a remarkable simple proof of the Gauss’s generalization of the Wilson’s theorem is given. The proof is based on properties of a subgroup generated by element of order 2 of a finite abelian group. Some conditions equivalent to the cyclicity of (Φ(n), ·n), where n > 2 is an integer are presented, in particular, a condition for the existence of the unique element of order 2 in such a group.

Small-sum pairs in abelian groups

Reza Akhtar, Paul Larson (2010)

Journal de Théorie des Nombres de Bordeaux

Let G be an abelian group and A , B two subsets of equal size k such that A + B and A + A both have size 2 k - 1 . Answering a question of Bihani and Jin, we prove that if A + B is aperiodic or if there exist elements a A and b B such that a + b has a unique expression as an element of A + B and a + a has a unique expression as an element of A + A , then A is a translate of B . We also give an explicit description of the various counterexamples which arise when neither condition holds.

Smooth invariants and ω -graded modules over k [ X ]

Fred Richman (2000)

Commentationes Mathematicae Universitatis Carolinae

It is shown that every ω -graded module over k [ X ] is a direct sum of cyclics. The invariants for such modules are exactly the smooth invariants of valuated abelian p -groups.

Solitary quotients of finite groups

Marius Tărnăuceanu (2012)

Open Mathematics

We introduce and study the lattice of normal subgroups of a group G that determine solitary quotients. It is closely connected to the well-known lattice of solitary subgroups of G, see [Kaplan G., Levy D., Solitary subgroups, Comm. Algebra, 2009, 37(6), 1873–1883]. A precise description of this lattice is given for some particular classes of finite groups.

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