#### Glasnik Matematicki, Vol. 42, No.2 (2007), 345-355.

### CYCLIC SUBGROUPS OF ORDER 4 IN FINITE 2-GROUPS

### Zvonimir Janko

Mathematical Institute, University of Heidelberg, 69120 Heidelberg, Germany

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

**Abstract.** We determine completely the structure of finite 2-groups
which possess exactly six cyclic subgroups of order 4.
This is an exceptional case because in a finite 2-group
is the number of cyclic subgroups of a given
order 2^{n} (*n* ≥ 2 fixed) divisible by 4 in most cases
and this solves a part of a problem stated by Berkovich.
In addition, we show that if in a finite 2-group *G*
all cyclic subgroups of order $4$ are conjugate, then *G* is cyclic or dihedral.
This solves a problem stated by Berkovich.

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

**Key words and phrases.** Finite 2-groups, 2-groups of maximal class,
minimal nonabelian 2-groups, *L*_{2}-groups, *U*_{2}-groups.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.42.2.08

**References:**

- Y. Berkovich,
*On the number of elements of given order in a finite **p*-group,
Israel J. Math. **73** (1991), 107-112.

MathSciNet

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

- 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,
*Finite 2-groups
**G* with |Ω_{2}(*G*)| = 16,
Glas. Mat. Ser. III **40** (2005), 71-86.

MathSciNet
CrossRef

- Z. Janko,
*Elements of order at most 4 in finite 2-groups, 2*,
J. Group Theory **8** (2005), 683-686.

MathSciNet
CrossRef

- Z. Janko,
*Finite 2-groups with exactly one nonmetacyclic maximal subgroup*,
submitted.

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