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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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.
For the second one it gets better: A group can have all its subgroups abelian without being abelian itself (Quaternion group)
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!
Originally Posted by Bruno 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. So, If a group is abelian, then , and so this definitely holds for any subset is a proof? Thanks both!
Originally Posted by MaryB So, If a group is abelian, then , and so this definitely holds for any subset is a proof? Thanks both! What do you think? Does it convince you?
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):
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