 These two definitions are equivalent, since for every group "H" and every normal subgroup "N" of "H", the quotient "H"/"N" is abelian if and only if "N" includes the commutator subgroup of "H".The least "n" such that "G"("n") = 1 is called the derived length of the solvable group "G".For finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order.