This is a group theory question. I am preparing a research paper. One result brought my attention. I am wondering if you know some paper or book listed this result.

Let $G$ be a 2-group. Suppose there exists nonabelian subgroups of $G$(FYI, otherwise the structure of $G$ is known, see Huppert, Endliche Gruppen I, P309). If every nonabelian subgroup is a hamiltonian 2-group, then $G$ is hamiltonian 2-group.

If no book list this results, Do you think it is an interesting result or a tedious one?

Thank you very much in advance for your comment.

Peter Tan

  • $\begingroup$ Do you mean every PROPER non-Abelian subgroup? $\endgroup$ – Geoff Robinson Mar 20 '14 at 8:44
  • $\begingroup$ Yes. proper subgroups. $\endgroup$ – Yilan Tan Mar 21 '14 at 9:12

Groups with this property were studied by G.A. Miller in 1907: see here.

It looks as though there is just one counterexample, the quaternion group of order $16$.


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