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.