Glasnik matematički, Vol. 43 No. 1, 2008.
Original scientific paper
https://doi.org/10.3336/gm.43.1.08
Some peculiar minimal situations by finite p-groups
Zvonimir Janko
; Mathematical Institute, University of Heidelberg
Abstract
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].
Keywords
Minimal nonabelian p-groups; metacyclic p-groups; 2-groups of maximal class; central products; Hamiltonian 2-groups
Hrčak ID:
23535
URI
Publication date:
25.5.2008.
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