#### Glasnik Matematicki, Vol. 47, No. 2 (2012), 325-332.

### FINITE *p*-GROUPS ALL OF WHOSE MAXIMAL SUBGROUPS, EXCEPT ONE, HAVE ITS DERIVED SUBGROUP OF ORDER *≤ p*

### Zvonimir Janko

Mathematical Institute, University of Heidelberg ,
69120 Heidelberg, Germany

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

**Abstract.** Let *G* be a finite *p*-group which has exactly one maximal subgroup *H* such that *|H'|>p*. Then we have d*(G)=2*, *p=2*, *H'* is a four-group, *G'* is abelian of order *8* and type *(4,2)*, *G* is of class *3* and the structure of *G* is completely determined.
This solves the problem Nr. 1800 stated by Y. Berkovich in [3].

**2010 Mathematics Subject Classification.**
20D15.

**Key words and phrases.** Finite *p*-groups, minimal nonabelian *p*-groups, commutator subgroups, nilpotence class of *p*-groups, Frattini subgroups,
generators and relations.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.47.2.08

**References:**

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

MathSciNet
CrossRef

- 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,
Walter de Gruyter, Berlin-New York, 2011.

MathSciNet
CrossRef

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