The equivalence of ×^{𝑡}𝐶≈×^{𝑡}𝐷 and 𝐽×𝐶≈𝐽×𝐷

Abstract

Let C satisfy the maximal condition for normal subgroups and let × t C ≈ × t D \times {\,^t}C\, \approx \, \times {\,^t}D for some positive integer t. Then C × J ≈ D × J C\, \times \,J\, \approx \,D\, \times \,J where J is the infinite cyclic group. If × s C ≈ × t D \times {\,^s}C\, \approx \, \times {\,^t}D and s ⩾ t s \geqslant \,t , there exists a finitely generated free abelian group S such that C is a direct factor of D × S D\, \times \,S .