  In this paper we investigate whether the stellar initial mass function of
early-type galaxies depends on their host environment. To this purpose, we have
selected a sample of early-type galaxies from the SPIDER catalogue,
characterized their environment through the group catalogue of Wang et al. and
used their optical SDSS spectra to constrain the IMF slope, through the
analysis of IMF-sensitive spectral indices. To reach a high enough
signal-to-noise ratio, we have stacked spectra in velocity dispersion
($\sigma_0$) bins, on top of separating the sample by galaxy hierarchy and host
halo mass, as proxies for galaxy environment. In order to constrain the IMF, we
have compared observed line strengths to predictions of MIUSCAT/EMILES
synthetic stellar population models, with varying age, metallicity, and
"bimodal" (low-mass tapered) IMF slope ($\rm \Gamma_b$). Consistent with
previous studies, we find that $\rm \Gamma_b$ increases with $\sigma_0$,
becoming bottom-heavy (i.e. an excess of low-mass stars with respect to the
Milky-Way-like IMF) at high $\sigma_0$. We find that this result is robust
against the set of isochrones used in the stellar population models, as well as
the way the effect of elemental abundance ratios is taken into account. We thus
conclude that it is possible to use currently state-of-the-art stellar
population models and intermediate resolution spectra to consistently probe IMF
variations. For the first time, we show that there is no dependence of
$\Gamma_b$ on environment or galaxy hierarchy, as measured within the $3"$ SDSS
fibre, thus leaving the IMF as an intrinsic galaxy property, possibly set
already at high redshift.
