Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For any abelian group $G$, there is a torsion subgroup $TG=\{g\in G, ng=0 \textrm{ for some non-zero integer} n\}$.

Now, let $A\to B\to C$ be an exact sequence of abelian groups.

Is it true that $TA\to TB\to C$ is exact? (every maps are restriction of given maps)

share|cite|improve this question

No. Consider the exact sequence $\mathbb Q \to \mathbb Q/\mathbb Z \to 0$. This restricts to $0 \to \mathbb Q/\mathbb Z \to 0,$ which is no longer exact.

Another (closely related) example is given by $\mathbb Z \to \mathbb Z/n\mathbb Z \to 0,$ for any $n > 1$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.