Groups Factorized by Pairwise Permutable Abelian Subgroups of Finite Rank

Abstract

It is proved that a group which is the product of pairwise permutable abelian subgroups of finite Prüfer rank is hyperabelian with finite Prüfer rank; in the periodic case the Sylow subgroups of such a product are described. Furthermore, if $G = ABC$ is such a non-periodic product with locally cyclic subgroups A, B and C, then the Prüfer rank of $G$ is at most $8$. Moreover, $G$ is soluble of derived length at most $4$ and has Prüfer rank at most 6, if $A\cap B\cap C = 1$, and $G$ has a torsion subgroup $T$ such that the factor group $G/T$ is locally cyclic and the Sylow $p$-subgroups of $T$ are of Prüfer rank at most $2$ for odd $p$ and at most $6$ for $p = 2$, otherwise.

Keywords

• Factorized group
• finite Prüfer rank
• locally cyclic group