An extension for Hajós' theorem
\textit{G. Hajós}' theorem asserts that if a finite abelian group is expressed as a direct product of cyclic subsets of prime cardinality, then at least one of the factors must be a subgroup [see Math. Z. 47, 427-467 (1941; Zbl 0025.25401)]. \textit{A. D. Sands} proved that in a factorization of a finite cyclic group by subsets of cardinalities that are prime powers at least one of the factors is periodic. [Proc. Colloq. Abelian Groups, 139-146 (1964; Zbl 0132.271)]. In the reviewed paper the authors prove the following Theorem 5: Let \(G\) be a finite abelian group and let \(R\) be a subset of the prime divisors of \(|G|\). Suppose that the \(r\)-components of \(G\) are cyclic for each \(r \in R\). If \(G = A_ 1\cdots A_ n\) is a normed factorization of \(G\), where for each \(i\), \(1 \leq i\leq n\), either \(A_ i\) is a pseudocyclic subset and hence of prime order or \(|A_i|\) is a power of one of the primes \(r \in R\), then one of the factors \(A_ i\) is periodic. The authors also show that pseudocyclicity is a generalization of cyclicity, and that if a normed subset of prime cardinality of \(G\) is periodic, then it is a subgroup. Hence, the theorem is a common generalization of Hajós' and Sands' theorem.
scientific article; zbMATH DE number 61224