in this paper we provide explicit formulas for the number of elements/subgroups/cyclic subgroups of a given order and for the total number of subgroups/cyclic subgroups in a finite hamiltonian group. the coverings with three proper subgroups and the principal series of such a group are also counted. finally, we give a complete description of the lattice of characteristic subgroups of a finite h...

In this paper we provide explicit formulas for the number of elements/subgroups/cyclic subgroups of a given order and for the total number of subgroups/cyclic subgroups in a finite Hamiltonian group. The coverings with three proper subgroups and the principal series of such a group are also counted. Finally, we give a complete description of the lattice of characteristic subgroups of a finite H...

A finite group is said to have "perfect order classes" if the number of elements any given either zero or a divisor group. The purpose this note describe explicitly Hamiltonian groups with perfect classes. We show that has classes if, and only it isomorphic direct product quaternion $8$, non-trivial cyclic $3$-group at most $2$. Theorem. $Q\times C_{3^k}$ C_{2}\times C_{3^k}$, for some positive...