With targeted imaging of groups in the local volume , the regions of collapse around bright galaxies can be clearly identified by the distribution of dwarfs and luminosity functions can be established to very faint levels .
In the case of the M81 Group there is completion to M _ { R } \sim - 9 .
In all well studied cases , the faint end slopes are in the range -1.35 < \alpha < -1.2 , much flatter than the slope for the bottom end of the halo mass spectrum anticipated by \Lambda CDM hierarchical clustering theory .
Small but significant variations are found with environment .
Interestingly , the populations of dwarf galaxies are roughly constant per unit halo mass .
With the numbers of dwarfs as an anchor point , evolved environments ( dominated by early morphological types ) have relatively fewer intermediate luminosity systems and at least one relatively more important galaxy at the core .
The variations with environment are consistent with a scenario of galaxy merging .
However it is questionable if the universal dearth of visible dwarf systems is a consequence of an astrophysical process like reionization .