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https://doi.org/10.3336/gm.43.1.08

Some peculiar minimal situations by finite p-groups

Zvonimir Janko ; Mathematical Institute, University of Heidelberg

Puni tekst: engleski, pdf (137 KB) str. 111-120 preuzimanja: 308* citiraj
APA 6th Edition
Janko, Z. (2008). Some peculiar minimal situations by finite p-groups. Glasnik matematički, 43 (1), 111-120. https://doi.org/10.3336/gm.43.1.08
MLA 8th Edition
Janko, Zvonimir. "Some peculiar minimal situations by finite p-groups." Glasnik matematički, vol. 43, br. 1, 2008, str. 111-120. https://doi.org/10.3336/gm.43.1.08. Citirano 24.09.2021.
Chicago 17th Edition
Janko, Zvonimir. "Some peculiar minimal situations by finite p-groups." Glasnik matematički 43, br. 1 (2008): 111-120. https://doi.org/10.3336/gm.43.1.08
Harvard
Janko, Z. (2008). 'Some peculiar minimal situations by finite p-groups', Glasnik matematički, 43(1), str. 111-120. https://doi.org/10.3336/gm.43.1.08
Vancouver
Janko Z. Some peculiar minimal situations by finite p-groups. Glasnik matematički [Internet]. 2008 [pristupljeno 24.09.2021.];43(1):111-120. https://doi.org/10.3336/gm.43.1.08
IEEE
Z. Janko, "Some peculiar minimal situations by finite p-groups", Glasnik matematički, vol.43, br. 1, str. 111-120, 2008. [Online]. https://doi.org/10.3336/gm.43.1.08

Sažetak
In this paper we show that a finite p-group which possesses non-normal subgroups and such that any two non-normal subgroups of the same order are conjugate must be isomorphic to

Mpn = < a,b | apn-1 = bp = 1, n ≥ 3, ab = a1+pn-2 >,

where in case p = 2 we must have n ≥ 4. This solves Problem Nr. 1261 stated by Y. Berkovich in [1]. In a similar way we solve Problem Nr. 1582 from [1] by showing that Mpn is the only finite p-group with exactly one conjugate class of non-normal cyclic subgroups.
Then we determine up to isomorphism all finite p-groups which possess non-normal subgroups and such that the normal closure HG of each non-normal subgroup H of G is the largest possible, i.e., |G : HG| = p. It turns out that G is either the nonabelian group of order p3, p > 2, and exponent p or G is metacyclic. This solves the Problem Nr. 1164 stated by Berkovich [1].

We classify also finite 2-groups with exactly two conjugate classes of four-subgroups. As a result, we get three classes of such 2-groups. This solves Problem Nr. 1260 stated by Y.Berkovich in [1].

Ključne riječi
Minimal nonabelian p-groups; metacyclic p-groups; 2-groups of maximal class; central products; Hamiltonian 2-groups

Hrčak ID: 23535

URI
https://hrcak.srce.hr/23535

Posjeta: 547 *