  Let G be a group and DS(G) = { H'| H is any subgroup of G}. G is said to be a
DC-group if DS(G) is a chain. In this paper, we prove that a finite DC-group is
a semidirect product of a Sylow p-subgroup and an abelian p'-subgroup. For the
case of G being a finite p-group, we obtain some properties of a DC-group. In
particular, a DC 2-group is characterized. Moreover, we prove that DC-groups
are metabelian for p<5 and give an example that a non-abelian DC-group is not
be necessarily metabelian for p>3.
