Finite 2-groups whose nonnormal subgroups have orders at most 2³

Abstract

In this paper, we classify finite 2-groups all of whose nonnormal subgroups have orders at most 2³. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.

Keywords

Minimal non-abelian p-group / nonnormal subgroup / central extension

References

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[2] Berkovich Y, Janko Z. Groups of Prime Power Order, Vol. 2. Berlin, New York: Walter de Gruyter, 2008

[3] Berkovich Y, Janko Z. Groups of Prime Power Order, Vol. 3. Berlin, New York: Walter de Gruyter, 2011

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[7] Zhang Q H, Li X X, Su M J. Finite p-groups whose nonnormal subgroups have orders≤p³ (in preparation)

[8] Zhang Q H, Sun X J, An L J, Xu M Y. Finite p-groups all of whose subgroups of index p² are abelian. Algebra Colloq, 2008, 15(1): 167-180

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg