# Thread: Is the subgroup of an abelian group always abelian?

1. ## Is the subgroup of an abelian group always abelian?

That's my question. I think it is, but can you proof this?

And is it both ways: and if a group has a abelian subgroup, is the group itself always abelian?

2. Yes, no.

Think about it! If a group is abelian, then $xy=yx \forall x,y \in G$, and so this definitely holds for any subset $S \subset G$.

On the other hand, if you take any group $G$, then the trivial subgroup $\{1\}$ is abelian, but $G$ isn't necessarily.

3. For the second one it gets better: A group can have all its subgroups abelian without being abelian itself (Quaternion group)

4. Originally Posted by Jose27
For the second one it gets better: A group can have all its (proper) subgroups abelian without being abelian itself (Quaternion group)
Yes, much better example!

5. Originally Posted by Bruno J.
Yes, no.

Think about it! If a group is abelian, then $xy=yx \forall x,y \in G$, and so this definitely holds for any subset $S \subset G$.

On the other hand, if you take any group $G$, then the trivial subgroup $\{1\}$ is abelian, but $G$ isn't necessarily.
So, If a group is abelian, then $xy=yx \forall x,y \in G$, and so this definitely holds for any subset $S \subset G$ is a proof?
Thanks both!

6. Originally Posted by MaryB
So, If a group is abelian, then $xy=yx \forall x,y \in G$, and so this definitely holds for any subset $S \subset G$ is a proof?
Thanks both!
What do you think? Does it convince you?

7. Originally Posted by MaryB
That's my question. I think it is, but can you proof this?
Also, if a group is abelian then so are all its homomorphic images (equivalently, quotients):

$(a\phi) (b \phi) = (ab)\phi = (ba)\phi = (b\phi) (a \phi)$