#### Glasnik Matematicki, Vol. 43, No.1 (2008), 111-120.

### SOME PECULIAR MINIMAL SITUATIONS BY FINITE *p*-GROUPS

### 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

*M*_{pn} =
< *a,b* | *a*^{pn-1} =
*b*^{p} = 1, *n* ≥ 3, *a*^{b} =
*a*^{1+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
*M*_{pn} 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 *H*^{G} of each non-normal subgroup *H* of *G* is the largest possible,
i.e., |*G* : *H*^{G}| = *p*. It turns out that *G*
is either the nonabelian group of order *p*^{3}, *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:**

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

- Y. Berkovich,
*Short proofs of
some basic characterization theorems of finite **p*-group theory,
Glas. Mat. Ser. III **41(61)** (2006), 239-258.

MathSciNet
CrossRef

- Y. Berkovich and Z. Janko,
*Structure of finite **p*-groups with given subgroups,
Contemporary Math. **402** (2006), 13-93.

MathSciNet

- Z. Janko,
*Finite 2-groups with small centralizer of an involution*,
J. of Algebra **241** (2001), 818-826.

MathSciNet
CrossRef

- Z. Janko,
*Finite 2-groups with small centralizer of an involution 2*,
J. of Algebra **245** (2001), 413-429.

MathSciNet
CrossRef

- Z. Janko,
*A classification of finite 2-groups with exactly three involutions*,
J. of Algebra **291** (2005), 505-533.

MathSciNet
CrossRef

- Z. Janko,
*Finite **p*-groups
with a uniqueness condition for non-normal subgroups,
Glas. Mat. Ser. III **40** (2007), 235-240.

MathSciNet
CrossRef

- Z. Janko,
*On finite nonabelian 2-groups all of whose minimal nonabelian subgroups are of exponent 4*,
J. Algebra **315** (2007), 801-808.

MathSciNet
CrossRef

- B. Huppert,
Endliche Gruppen 1, Springer, Berlin, 1967.

MathSciNet

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