#### Glasnik Matematicki, Vol. 40, No.1 (2005), 71-86.

### FINITE 2-GROUPS *G* WITH
|Ω_{2}(*G*)| = 16

### Zvonimir Janko

Mathematical Institute, University of Heidelberg,
69120 Heidelberg, Germany

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

**Abstract.** It is a known fact that the subgroup
Ω_{2}(*G*)
generated by all elements of order at most 4 in a finite 2-group
*G* has a strong influence on the structure of the whole group.
Here we determine finite 2-groups *G* with
|*G*| > 16 and
Ω_{2}(*G*) = 16.
The resulting groups are only in one case metacyclic
and we get in addition eight infinite classes of non-metacyclic
2-groups and one exceptional group of order 2^{5}. All
non-metacyclic 2-groups will be given in terms of generators and
relations.

In addition we determine completely finite 2-groups *G*
which possess exactly one abelian subgroup of type (4,2).

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

**Key words and phrases.** 2-group, metacyclic group,
Frattini subgroup, self-centralizing subgroup.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.40.1.08

**References:**

- Y. Berkovich,
*On subgroups and epimorphic images of finite **p*-groups,
J. Algebra **248** (2002), 472-553.

- Y. Berkovich,
Selected topics of finite group theory, in preparation.

- Z. Janko,
*Finite 2-groups with small centralizer of an involution, 2*,
J. Algebra, **245** (2001), 413-429.

CrossRef

- Z. Janko,
*Finite 2-groups with no normal elementary abelian subgroups of order 8*,
J. Algebra **246** (2001), 951-961.

CrossRef

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

CrossRef

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