Do the stellar populations of the brightest two group galaxies depend on
  the magnitude gap?
  We investigate how the stellar populations of the inner regions of the first
and the second brightest group galaxies (respectively BGGs and SBGGs) vary as a
function of magnitude gap, using an SDSS-based sample of 550 groups with
elliptical BGGs. The sample is complete in redshift, luminosity and for $\Delta
\mathcal{M}_{12}$ up to 2.5 mag, and contains 59 large-gap groups (LGGs, with
$\Delta \mathcal{M}_{12} > 2.0$ mag). We determine ages, metallicities, and
SFHs of BGGs and SBGGs using the STARLIGHT code with two different single
stellar population models (which lead to important disagreements in SFHs), and
also compute $[\alpha/{\rm Fe}]$ from spectral indices. After removing the
dependence with galaxy velocity dispersion or with stellar mass, there is no
correlation with magnitude gap of BGG ages, metallicities, $[\alpha/{\rm Fe}]$,
and SFHs. The lack of trends of BGG SFHs with magnitude gap suggests that BGGs
in LGGs have undergone more mergers than those in small-gap groups, but these
mergers are either dry or occurred at very high redshift, which in either case
would leave no detectable imprint in their spectra. We show that SBGGs in LGGs
lie significantly closer to the BGGs (in projection) than galaxies with similar
stellar masses in normal groups, which appears to be a sign of the earlier
entry of the former into their groups. Nevertheless, the stellar population
properties of the SBGGs in LGGs are compatible with those of the general
population of galaxies with similar stellar masses residing in normal groups.
