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?

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- December 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? - December 4th 2009, 10:45 AMBruno J.
Yes, no.

Think about it! If a group is abelian, then , and so this definitely holds for any subset .

On the other hand, if you take any group , then the trivial subgroup is abelian, but isn't necessarily. - December 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)

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