Subgroups of $\mathbb{Z}^n$ of rank $n$

Question

I found from here that groups $A \supset B \supset C$ with $A \simeq C$ does not imply $A\simeq B$. How about the case of $A\simeq \mathbb{Z}^n$? Is it still false?

score 2 · Accepted Answer · answered May 05 '13 at 13:20

2

Subgroups of free abelian groups are free abelian and have smaller (or equal) rank.

answered May 05 '13 at 13:20

Cameron Buie · Answer 2 · 2013-05-05T13:33:27.270

1

In that special case, it is true. In particular, $B$ will necessarily be a free abelian group of rank $n$.

edited May 05 '13 at 13:33

answered May 05 '13 at 13:19

Cameron Buie

105,149

Subgroups of $\mathbb{Z}^n$ of rank $n$

2 Answers2