What is the smallest order for a group to have a non abelian proper subgroups? Is there any efficient method to answer this question.

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1 Answer

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You want to minimize the index [G:H][G:H], so the smallest size for |G||G| can be 2|H|2 |H|. This situation exists, for example H×Z2H \times \mathbb{Z}_2. The smallest non abelian group is S3S_3, so the minimal size of a group with non abelian subgroup is 1212, like S3×Z2S_3 \times \mathbb{Z}_2.

Then it should be S4S_4 as S3S_3 has all subgroups as abelian one.

– vikrant

2 days ago

You misunderstand, S3⊂S3×Z2S_3 \subset S_3 \times \mathbb{Z}_2 is not abelian.

– basket

2 days ago