Frontiers of Mathematics in China >
Finite p-groups whose non-normal subgroups have few orders
Received date: 21 Dec 2017
Accepted date: 26 Feb 2018
Published date: 14 Aug 2018
Copyright
Suppose that G is a nite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use and to denote the maximum and minimum of the orders of the non-normal subgroups of G; respectively. In this paper, we classify groups G such that : As a by-product, we also classify p-groups whose orders of non-normal subgroups are and :
Key words: Finite p-groups; meta-hamiltonian p-groups; non-normal subgroups
Lijian AN . Finite p-groups whose non-normal subgroups have few orders[J]. Frontiers of Mathematics in China, 2018 , 13(4) : 763 -777 . DOI: 10.1007/s11464-018-0693-0
1 |
An L, Li L, Qu H, Zhang Q. Finite p-groups with a minimal non-abelian subgroup of index p (II). Sci China Math, 2014, 57: 737–753
|
2 |
An L, Zhang Q. Finite metahamiltonian p-groups. J Algebra, 2015, 442: 23–35
|
3 |
Berkovich Y. Groups of Prime Power Order, Vol. 1. Berlin: Walter de Gruyter, 2008
|
4 |
Fang X, An L. The classication of nite metahamiltonian p-groups. arXiv.org: 1310.5509v2
|
5 |
Passman D S. Nonnormal subgroups of p-groups. J Algebra, 1970, 15: 352–370
|
6 |
Rédei L. Das “schiefe Produkt” in der Gruppentheorie mit Anwendung auf die endlichen nichtkommutativen Gruppen mit lauter kommutativen echten Untergruppen und die Ordnungszahlen, zu denen nur kommutative Gruppen gehören (German). Comment Math Helvet, 1947, 20: 225–264
|
7 |
Xu M, An L, Zhang Q. Finite p-groups all of whose non-abelian proper subgroups are generated by two elements. J Algebra, 2008, 319: 3603–3620
|
8 |
Zhang Q, Guo X, Qu H, Xu M. Finite group which have many normal subgroups. J Korean Math Soc, 2009, 46(6): 1165–1178
|
9 |
Zhang Q, Li X, Su M. Finite p-groups whose nonnormal subgroups have orders at most p3: Front Math China, 2014, 9(5): 1169–1194
|
10 |
Zhang Q, Su M. Finite 2-groups whose nonnormal subgroups have orders at most 23: Front Math China, 2012, 7(5): 971–1003
|
11 |
Zhang Q, Zhao L, Li M, Shen Y. Finite p-groups all of whose subgroups of index p3 are abelian. Commun Math Stat, 2015, 3: 69–162
|
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