#### Glasnik Matematicki, Vol. 41, No.1 (2006), 71-76.

### FINITE 2-GROUPS *G* WITH Ω_{2}*(*G*)
METACYCLIC

### Zvonimir Janko

Mathematical Institute, University of Heidelberg,
69120 Heidelberg, Germany

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

**Abstract.** In this paper we classify finite non-metacyclic 2-groups
*G* such that Ω_{2}*(*G*) (the subgroup generated by all
elements of order 4) is metacyclic. However, if *G* is a finite
2-group such that Ω_{2}(*G*) (the subgroup generated by all
elements of order ≤ 4) is metacyclic, then *G* is
metacyclic.

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

**Key words and phrases.** Finite 2-groups, 2-groups of maximal class, metacyclic
groups.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.41.1.07

**References:**

- Z. Janko,
*Finite 2-groups with a self-centralizing elementary abelian subgroup of order 8*,
J. Algebra **269** (2003), 189-214.

MathSciNet
CrossRef

- Z. Janko,
*Finite 2-groups with exactly four cyclic subgroups of order 2*^{n},
J. Reine Angew. Math. **566** (2004), 135-181.

MathSciNet
CrossRef

- Z. Janko,
*Elements of order at most 4 in finite 2-groups*,
J. Group Theory **7** (2004), 431-436.

MathSciNet
CrossRef

- Z. Janko,
*Finite 2-groups **G* with
|Ω_{2}(*G*)| = 16,
Glasnik Mat. **40(60)** (2005), 71-86.

MathSciNet
CrossRef

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

MathSciNet
CrossRef

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