Abstract
We show that if all nonnormal subgroups of a non-Dedekindian p-group G are elementary abelian, then |G′| = p, unless G = D16 is dihedral of order 16. We also describe the p-groups of exponent > p > 2 all of whose nonnormal subgroups have exponent p. We also present a new proof of Passman’s theorem [Ber1, Theorem 1.25] classifying the non-Dedekindian p-groups all of whose nonnormal subgroups have order p. A number of related problems is stated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Y. Berkovich, Groups of Prime Power Order, Volume 1, de Gruyter Expositions in Mathematics, Vol. 46, W. de Gruyter, Berlin, 2008.
Y. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 2, de Gruyter Expositions in Mathematics, Vol. 47, W. de Gruyter, Berlin, 2008.
X. G. Fang and L. J. An, A classification of finite metahamiltonian p-groups, arXiv: 1310.5509v1 [math.GR]_21 Oct 2013.
A. Mann, Finite groups with maximal normalizers, Illinois Journal of Mathematics 12 (1968), 67–75.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Berkovich, Y. On the theorem of Mann about p-groups all of whose nonnormal subgroups are elementary abelian. Isr. J. Math. 208, 451–460 (2015). https://doi.org/10.1007/s11856-015-1207-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-015-1207-3