Example of non abelian group in which all its subgroups are normal
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The symmetric group is an example of a finite non-abelian group in which every proper subgroup is abelian. This group is not simple because its Sylow 3-subgroup is normal. In this post, we'll show that this is the case for any finite (non-abelian) group all of whose proper subgroups are abelian.
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Quaternion group which is non abelian but it's all subgroups are normal
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