The isomorphism of quotient groups

Question

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 $X/B$ ?

No. There are examples in which $X, A, B$ are all isomorphic to $\mathbb{Z}$. — Qiaochu Yuan, Jul 27 '12 at 03:50

score 2 · Accepted Answer · answered Jul 27 '12 at 04:00

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.

The isomorphism of quotient groups

1 Answers1