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All subgroups of a abelian group are normal. But the converse is not true. If every subgroup of a group is normal, then what more can we say about the group?

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en.wikipedia.org/wiki/Dedekind_group –  anon Jun 28 '12 at 10:15
    
Thank you for reference. –  DurgaDatta Jun 28 '12 at 10:19

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up vote 11 down vote accepted

If $G$ is a finite non-abelian group where all subgroups are normal, then $$G \cong Q_8 \times A \times B$$ where $A$ is an elementary abelian 2-group (ie, all non-identity elements have order 2), $B$ is abelian of odd order and $Q_8$ is the quaternion group of order 8. A proof can be found in for example Berkovich's Groups of Prime Power Order I believe.

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Is something known about infinite groups? –  user23211 Jun 28 '12 at 10:34
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Groups where all subgroups are normal are called Dedekind groups. The finite case as explained above is proved in M. Hall, Theory of Groups. –  i. m. soloveichik Jun 28 '12 at 12:52
    
What does "elementary" mean in the phrase "elementary abelian 2-group"? –  MJD Jun 28 '12 at 12:59
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An elementary abelian group is one where all elements (except the identity) have the same order. This order is then necessarily a prime. –  Tobias Kildetoft Jun 28 '12 at 13:12

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