#### Glasnik Matematicki, Vol. 41, No.2 (2006), 271-274.

### FINITE NONABELIAN 2-GROUPS IN WHICH ANY TWO NONCOMMUTING ELEMENTS GENERATE A
SUBGROUP OF MAXIMAL CLASS

### Zvonimir Janko

Mathematical Institute, University of Heidelberg, 69120 Heidelberg, Germany

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

**Abstract.**
We determine here the structure of the title groups.
It turns out that such a group *G*
is either quasidihedral or *G* = *HZ*(*G*),
where *H* is of maximal
class or extraspecial and
(*Z*(*G*)) ≤ *Z*(*H*).
This solves a
problem stated by Berkovich. The corresponding problem for *p* > 2
is open but very difficult since the *p*-groups of maximal class
are not classified for *p* > 2.

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

**Key words and phrases.** Finite 2-groups, 2-groups of
maximal class, minimal nonabelian 2-groups, quasidihedral
2-groups, Hughes *H*_{p}-subgroups.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.41.2.09

**References:**

- Y. Berkovich,
*Groups of prime power order, Parts I, II, and III* (with Z. Janko), in preparation.

- L. S. Kazarin,
*Groups with certain conditions imposed on the normalizers of
subgroups*, Perm. Gos. Univ. Ucen. Zap. **218** (1969),
268-279 (Russian).

MathSciNet

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

MathSciNet
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