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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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    http://en.wikipedia.org/wiki/Dedekind_group2012-06-28

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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?2012-06-28
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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.2012-06-28
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    What does "elementary" mean in the phrase "elementary abelian 2-group"?2012-06-28
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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.2012-06-28