### Zvonimir Janko

Mathematical Institute, University of Heidelberg, 69120 Heidelberg, Germany
e-mail: janko@mathi.uni-heidelberg.de

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].

2000 Mathematics Subject Classification.   20D15.

Key words and phrases.   Minimal nonabelian p-groups, metacyclic p-groups, 2-groups of maximal class, central products, Hamiltonian 2-groups.

Full text (PDF) (free access)

DOI: 10.3336/gm.43.1.08

References:

1. Y. Berkovich, Groups of prime power order, I and II (with Z. Janko), Walter de Gruyter, Berlin, 2008, to appear.

2. Y. Berkovich, Short proofs of some basic characterization theorems of finite p-group theory, Glas. Mat. Ser. III 41(61) (2006), 239-258.
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3. Y. Berkovich and Z. Janko, Structure of finite p-groups with given subgroups, Contemporary Math. 402 (2006), 13-93.
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4. Z. Janko, Finite 2-groups with small centralizer of an involution, J. of Algebra 241 (2001), 818-826.
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5. Z. Janko, Finite 2-groups with small centralizer of an involution 2, J. of Algebra 245 (2001), 413-429.
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6. Z. Janko, A classification of finite 2-groups with exactly three involutions, J. of Algebra 291 (2005), 505-533.
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7. Z. Janko, Finite p-groups with a uniqueness condition for non-normal subgroups, Glas. Mat. Ser. III 40 (2007), 235-240.
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8. Z. Janko, On finite nonabelian 2-groups all of whose minimal nonabelian subgroups are of exponent 4, J. Algebra 315 (2007), 801-808.
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9. B. Huppert, Endliche Gruppen 1, Springer, Berlin, 1967.
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