  We study the heavy quark symmetry with the homogeneous bag model (HBM) and
light-front quark model (LFQM) based on the decays of
$\Lambda_b^0\to\Lambda_c^+\ell^-\overline{\nu}_\ell~(\ell=e,\mu,\tau)$. In
particular, we calculate various parameters in the heavy quark expansions,
including the Isgur-Wise functions and their first order corrections. The
parameters in the HBM are fitted from the mass spectra, while the ones in the
LFQM are tightly constrained by the heavy quark symmetry, granting the
predictive power of our results. We explicitly obtain that ${\cal
B}(\Lambda_b^0\to\Lambda_c^+e^-\overline{\nu}_e)=(5.69\pm 0.58, 5.35\pm 0.50)$,
${\cal B}(\Lambda_b^0\to\Lambda_c^+\mu^-\overline{\nu}_\mu)=(5.67\pm 0.58 $,
$5.33\pm 0.49)$, and
$\Gamma(\Lambda_b^0\to\Lambda_c^+\tau^-\overline{\nu}_\tau)/\Gamma(\Lambda_b^0\to\Lambda_c^+\mu^-\overline{\nu}_\mu)=(0.3243\pm
0.0126$, $ 0.3506\pm 0.0046)$ for the numerical values of (HBM, LFQM). Our
results of the branching fractions in both models agree well with the
experimental data and lattice QCD calculations. In addition, we find that the
hard gluon corrections decrease the branching fractions around $10\%$.
