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

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.

DOI: 10.3336/gm.49.2.07

References:

1. Y. Berkovich, Groups of prime power order, Vol. 1, Walter de Gruyter, Berlin-New York, 2008.
   MathSciNet    

2. Y. Berkovich and Z. Janko, Groups of prime power order, Vol. 3, Walter de Gruyter, Berlin-New York, 2011.
   MathSciNet    

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