For a knot $K$ in $S^{3}$, J. Ma and R. Qiu defined an integral invariant $a(K)$ which is the minimal
number of elements that generate normally the commutator subgroup
of the knot group, and showed that it is a lower bound of
the unknotting number. We prove that it is also a lower bound
of the tunnel number. If the invariant were additive under
connected sum, then we could deduce something about additivity
of both the unknotting numbers and the tunnel numbers. However,
we found a sequence of examples that the invariant is not additive
under connected sum. Let $T(2, p)$ be the $(2, p)$-torus
knot, and $K_{p, q}=T(2, p) \sharp T(2, q)$.
Then we have $a(K_{p, q})=1$ if and only if $\gcd(p, q)= 1$.