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In [H. P. Tong-Viet, Simple classical groups of Lie type are determined by their character degrees, J. Algebra 357 (2012) 61-68] the following question arose: Question. Which groups can be uniquely determined by the structure of their complex group algebras? It is proved that every quasisimple group except covers of the alternating groups is uniquely determined up to isomorphism by the structure of , the complex group algebra of . One of the next natural groups to be considered are the characteristically simple groups. In this paper, as the first step in this investigation we prove that if is an odd prime number, then is uniquely determined by the structure of its complex group algebra. [ABSTRACT FROM AUTHOR]

Let B be a torsion-free finitely faithful S-group. This paper investigates the class of all torsion-free Abelian groups A with the property that Ext(A, B) is torsion-free. Our discussion is based on a natural extension of the concept of coseparability. [ABSTRACT FROM AUTHOR]

This paper investigates to which extent a self-small mixed Abelian group G of finite torsion-free rank is determined by the groups Hom(G,C) where C is chosen from a suitable class $\mathcal C$ of Abelian groups. We show that G is determined up to quasi-isomorphism if $\mathcal C$ is the class of all self-small mixed groups C with r0(C) ≤ r0(G). Several related results are given, and the dual problem of orthogonal classes is investigated. [ABSTRACT FROM AUTHOR]

Recently, Hoit introduced arithmetic on blocks, which extends the binary string operation by Jacobs and Keane. A string of elements from the Abelian additive group of residues modulo m, (Zm, ⊕), is called an m-block. The set of m-blocks together with Hoit's new product operation form an interesting algebraic structure where associative law and cancellation law hold. A weaker form of unique factorization and criteria for two indecomposable blocks to commute are also proved. In this paper, we extend Hoit's results by replacing the Abelian group (Zm, ⊕) by an arbitrary monoid (A, ◦). The set of strings built up from the alphabet A is denoted by String(A). We extend the operation ◦ on the alphabet set A to the string set String(A). We show that (String(A), ◦) is a monoid if and only if (A, ◦) is a monoid. When (A, ◦) is a group, we prove that stronger versions of a cancellation law and unique factorization hold for (String(A), ◦). A general criterion for two irreducible strings to commute is also presented. [ABSTRACT FROM AUTHOR]