#### Glasnik Matematicki, Vol. 45, No.1 (2010), 63-83.

### FINITE 2-GROUPS WITH EXACTLY ONE MAXIMAL
SUBGROUP WHICH IS NEITHER ABELIAN NOR MINIMAL NONABELIAN

### Zdravka Božikov and Zvonimir Janko

Faculty of Civil Engineering and Architecture, University of Split,
21000 Split, Croatia

*e-mail:* `Zdravka.Bozikov@gradst.hr`
Mathematical Institute, University of Heidelberg,
69120 Heidelberg, Germany

*e-mail:* `janko@mathi.uni-heidelberg.de`

**Abstract.** We shall determine the title groups *G* up to isomorphism.
This solves the problem Nr.861 for *p*=2 stated by Y. Berkovich in [2].
The resulting groups will be presented in terms of generators and relations. We
begin with the case d(*G*) = 2 and then we determine such groups for
d(*G*) > 2.
In these theorems we shall also describe all important characteristic subgroups so
that it will be clear that groups appearing in distinct
theorems are non-isomorphic.
Conversely, it is easy to check that all groups given in these theorems
possess exactly one maximal subgroup
which is neither abelian nor minimal nonabelian.

**2000 Mathematics Subject Classification.**
20D15.

**Key words and phrases.** Minimal nonabelian 2-groups, central products, metacyclic groups,
Frattini subgroups, generators and relations.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.45.1.06

**References:**

- Y. Berkovich,
Groups of prime power order, Vol. 1,
Walter de Gruyter, Berlin-New York, 2008.

MathSciNet

- Y. Berkovich and Z. Janko,
Groups of prime power order, Vol. 2,
Walter de Gruyter, Berlin-New York, 2008.

MathSciNet

- Y. Berkovich and Z. Janko,
Groups of prime power order, Vol. 3,
in preparation.

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