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?

Printable View

- Dec 4th 2009, 10:41 AMMaryBIs 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? - Dec 4th 2009, 10:45 AMBruno J.
Yes, no.

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

On the other hand, if you take any group $\displaystyle G$, then the trivial subgroup $\displaystyle \{1\}$ is abelian, but $\displaystyle G$ isn't necessarily. - Dec 4th 2009, 10:58 AMJose27
For the second one it gets better: A group can have all its subgroups abelian without being abelian itself (Quaternion group)

- Dec 4th 2009, 11:20 AMBruno J.
- Dec 6th 2009, 03:14 AMMaryB
- Dec 6th 2009, 10:06 AMBruno J.
- Dec 6th 2009, 11:38 PMSwlabr