  We use very high-S/N stacked spectra of $\sim$29,000 nearby quiescent
early-type galaxies (ETGs) from the Sloan Digital Sky Survey (SDSS) to
investigate variations in their star formation histories (SFHs) with
environment at fixed position along and perpendicular to the Fundamental Plane
(FP). We define three classifications of local group environment based on the
`identities' of galaxies within their dark matter halos: central `Brightest
Group Galaxies' (BGGs); Satellites; and Isolateds (those `most massive' in a
dark matter halo with no Satellites). We find that the SFHs of quiescent ETGs
are almost entirely determined by their structural parameters $\sigma$ and
$\Delta I_e$. Any variation with local group environment at fixed structure is
only slight: Satellites have the oldest stellar populations, 0.02 dex older
than BGGs and 0.04 dex older than Isolateds; BGGs have the highest
Fe-enrichments, 0.01 dex higher than Isolateds and 0.02 dex higher than
Satellites; there are no differences in Mg-enhancement between BGGs, Isolateds,
and Satellites. Our observation that, to zeroth-order, the SFHs of quiescent
ETGs are fully captured by their structures places important qualitative
constraints on the degree to which late-time evolutionary processes (those
which occur after a galaxy's initial formation and main star-forming lifetime)
can alter their SFHs/structures.

Astrophysics of Galaxies