#### Glasnik Matematicki, Vol. 49, No. 2 (2014), 333-336.

### FINITE *P*-GROUPS IN WHICH THE NORMAL CLOSURE OF EACH NON-NORMAL CYCLIC SUBGROUP IS NONABELIAN

### Zvonimir Janko

Mathematical Institute, University of Heidelberg,
69120 Heidelberg, Germany

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

**Abstract.**
We determine up to isomorphism finite non-Dedekindian *p*-groups *G* (i.e., *p*-groups which possess non-normal subgroups) such that the normal closure
of each non-normal cyclic subgroup in *G* is nonabelian. It turns out that we must have *p=2* and *G* has an abelian maximal subgroup *A*
of exponent *2*^{e}, *e≥ 3*, and an element *v G-A* such that for all *h A* we have either *h*^{v}=h^{-1} or *h*^{v}=h^{ -1+2e-1}.

**2010 Mathematics Subject Classification.**
20D15.

**Key words and phrases.** Finite *p*-groups, normal closure, quasidihedral *2*-groups, quasi-generalized quaternion groups, exponent of a *p*-group.

**Full text (PDF)** (access from subscribing institutions only)
DOI: 10.3336/gm.49.2.07

**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. 3, Walter de Gruyter, Berlin-New York, 2011.

MathSciNet

- Y. Berkovich and Z. Janko, Groups of prime power order, Vol. 5, Walter de Gruyter, Berlin-New York, 2014.

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