Direct decompositions of finite rank torsion-free Abelian groups

A. V. Yakovlev

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 8, Tome 160 (1987), pp. 272-285

Résumé

It is proved that if $r_1,r_2,\dots,r_s$; $l_1,l_2,\dots,l_t$ are the ranks of the indecomposable summands of two direct decompositions of a torsion-free Abelian group of finite rank and if $s_0$ is the number of units among the numbers $r_i$, while $t_0$ is the number of units among the numbers $l_j$, then $r_i\leq n-t_0$, $l_j\leq n-s_0$ for all $i$, $j$. Moreover, if for some i we have $i$ $r_i=n-t_0$, then among the $l_j$ only one term is different from 1 and it is equal to $n-t_0$; similarly if $l_j=n-s_0$ for some $j$. In addition, a construction is presented, allowing to form, from several indecomposable groups, a new group, called a flower group, and it is proved that a flower group is indecomposable under natural restrictions on its defining parameters.