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?
All subgroup are Normal
4
$\begingroup$
group-theory
-
5http://en.wikipedia.org/wiki/Dedekind_group – 2012-06-28
1 Answers
12
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.
-
1Is something known about infinite groups? – 2012-06-28
-
1Groups 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
-
0What does "elementary" mean in the phrase "elementary abelian 2-group"? – 2012-06-28
-
2An 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