#### Glasnik Matematicki, Vol. 46, No.1 (2011), 103-120.

### FINITE *p*-GROUPS *G* WITH *p* > 2 AND *d*(*G*) > 2 HAVING
EXACTLY ONE MAXIMAL SUBGROUP WHICH IS NEITHER ABELIAN NOR MINIMAL NONABELIAN

### Zvonimir Janko

Mathematical Institute,
University of Heidelberg,
69120 Heidelberg,
Germany

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

**Abstract.** We give here a complete classification (up to isomorphism) of the title groups (Theorems 1, 3 and 5). The corresponding problem for *p=2* was solved in [4] and for *p>2* with d*(G)=2 * was solved in [5]. This gives a complete solution of the problem Nr. 861 of Y. Berkovich stated in [2].

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

**Key words and phrases.** Minimal nonabelian *p*-groups, A_{2}-groups, metacyclic *p*-groups, Frattini subgroups, Hall-Petrescu formula, generators and relations,
congruences mod *p*.

**Full text (PDF)** (access from subscribing institutions only)
DOI: 10.3336/gm.46.1.11

**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,
Walter de Gruyter, Berlin-New York, to appear 2011.

- Z. Božikov and Z. Janko,
*Finite **2*-groups with exactly one maximal subgroup which is neither abelian nor minimal nonabelian,
Glas. Mat. Ser. III **45(65)** (2010), 63-83.

MathSciNet
CrossRef

- Z. Janko,
*Finite **p*-groups *G* with *p>2* and d*(G)=2* *having exactly one maximal subgroup which is neither abelian nor minimal nonabelian,*
Glas. Mat. Ser. III **45(65)** (2010), 441-452.

MathSciNet
CrossRef

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