  For a knot $K$ in $S^3$, let $G(K)$ be the knot group of $K$, $a(K)$ the
Ma-Qiu index (the MQ index, for short), which is the minimal number of normal
generators of the commutator subgroup of $G(K)$, and $m(K)$ the Nakanishi index
of $K$, which is the minimal number of generators of the Alexander module of
$K$.We generalize the notions for a pair of a group $G$ and its normal sugroup
$N$, and we denote them by $a(G, N)$ and $m(G, N)$ respectively.Then it is easy
to see $m(G, N)\le a(G, N)$ in general.We also introduce a notion
``$\omega$-solvability" for a group that the intersection of all higher
commutator subgroups is trivial.Our main theorem is that if $N$ is
$\omega$-solvable, then we have $m(G, N)=a(G, N)$.As corollaries, for a fibered
knot $K$, we have $m(K)=a(K)$, and we could determine the MQ indices of prime
knots up to $9$ crossings completely.
