  A theoretical research on possible effects of the Fierz transformations on
the ground states (vacua) of some 2-flavor and $N_c$-color four-fermion (quark)
interaction models has been systematically conducted. It has been shown that,
based on the known criterions of the interplay between the antiquark-quark
($\bar{q}$-$q$) and diquark ($q$-$q$) condensates, in 4D space-time, for the
given $\bar{q}$-$q$ channel couplings with chiral symmetry and from the heavy
gluon exchange, the effects of the Fierz transformations are not enough to
change the feature that the models' vacua could only be in the pure
$\bar{q}$-$q$ condensate phases. However, for a given pure scalar $q$-$q$
channel coupling with the strength $H_S$, the Fierz transformations will lead
to the nontrivial effect that the model's vacuum could be in the expected
$q$-$q$ condensate phase only if $N_c<9$ and $H_S$ is small, and as the
increase of $N_c$ and $H_S$, the vacuum will get first in a coexistence phase
with $q$-$q$ and $\bar{q}$-$q$ condensates then in a pure $\bar{q}$-$q$
condensate phase until $N_c\rightarrow\infty$. Similar conclusions are also
drawn from relevant four-fermion interaction models in 2D and 3D space-time.
The general significance of the research is indicated.
