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

If $X$ is an abelian group and $A,B$ are its subgroup with $A\cong B$, is quotient group $X/A$ isomorphic to the quotient group $X/B$ ?

share|cite|improve this question
No. There are examples in which $X, A, B$ are all isomorphic to $\mathbb{Z}$. – Qiaochu Yuan Jul 27 '12 at 3:50
See the answer to this question. – Zev Chonoles Jul 27 '12 at 3:53
up vote 2 down vote accepted

Let X be the group of integers under addition. Let A be the group of even integers, B the group of integers divisible by 3. Then X, A, and B are isomorphic to one another, they are all infinite cyclic groups. However X/A is the group of integers mod 2, a cyclic group of order 2, and X/B is the group of integers mod 3, a cyclic group of order 3.

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.