we know that if a group is abelian, all subgroups of it are normal. however - can you prove or disprove that if all subgroups of a group are normal, it is abelian?

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- Aug 24th 2010, 10:55 PMshos[SOLVED] Normal subgroups
we know that if a group is abelian, all subgroups of it are normal. however - can you prove or disprove that if all subgroups of a group are normal, it is abelian?

- Aug 24th 2010, 11:10 PMIsomorphism
We will disprove the claim.

Consider the group where . Also we need . Using the above relations you can show that is a group. You can also show it is non-abelian by observing that .

Note that is generated by any two of .

Now show that if any subgroup contains then . If , then index of in is 2 and thus is normal in . Suppose , then one of or must belong to . But then .

We can proceed similarly for the case where a subgroup contains or .

The only remaining case is that does not contain , or . In this case , which is the center of and thus is normal in