On the structure of generalized Hamiltonian groups

Abstract.

We consider the characteristic subgroup CS(G), generated by the nonnormal cyclic subgroups of the group G. A group G is called a generalized Dedekind group if $CS(G)\neq G$, and those among them with nontrivial CS(G) are called generalized Hamiltonian groups. Such groups are torsion groups of nilpotency class two. The commutator subgroup is cyclic of p-power or two times p-power order and always contained in CS(G). The quotient G/CS(G) is a locally cyclic p-group. We give an example of an infinite generalized Hamiltonian p-group with G/CS(G) locally cyclic.