  Using multi-band data we examine the star formation activity of the nearby
group-dominant early-type galaxies of the Complete Local-volume Groups Sample
(CLoGS), and the relation between star formation, gas content, and local
environment. Only a small fraction of the galaxies (13%; 6/47) are found to be
Far-Ultraviolet (FUV) bright, with FUV to near-infrared colours indicative of
recent active star formation (NGC 252, NGC 924, NGC 940, NGC 1106, NGC 7252,
and ESO 507-25). These systems are lenticulars presenting the highest FUV
specific star-formation rates in the sample (sSFR FUV > 5$\times$10$^{13}$
yr$^{-1}$), significant cold gas reservoirs (M(H2)=0.5 - 61$\times$10$^8$
M$_\odot$), reside in X-ray faint groups, and none of them hosts a powerful
radio AGN (P$_{1.4GHz}$<$10^{23}$ W Hz$^{-1}$). The majority of the
group-dominant galaxies (87%; 41/47) are FUV faint, with no significant star
formation, classified in most cases as spheroids based on their position on the
infrared star-forming main sequence (87%; 46/53). Examining the relationships
between radio power, SFR FUV, and stellar mass we find a lack of correlation
that suggests a combination of origins for the cool gas in these galaxies,
including stellar mass loss, cooling from the intra-group medium (IGrM) or
galaxy halo, and acquisition through mergers or tidal interactions. X-ray
bright systems, in addition to hosting radio powerful AGN, have a range of SFRs
but, with the exception of NGC 315, do not rise to the highest rates seen in
the FUV bright systems. We suggest that central group galaxy evolution is
linked to gas mass availability, with star formation favoured in the absence of
a group-scale X-ray halo, but AGN jet launching is more likely in systems with
a cooling IGrM.
