The satellite populations of the Milky Way, and Milky-Way-mass galaxies in the local universe, have been extensively studied to constrain dark-matter and galaxy-evolution physics. Recently, there has been a shift to studying satellites of hosts with stellar masses between that of the Large Magellanic Cloud and the Milky Way, since they can provide further insight on hierarchical structure formation, environmental effects on satellites, and the nature of dark-matter. Most work is focused on the Local Volume, and little is still known about low-mass host galaxies at higher red-shift. To improve our understanding of the evolution of satellite populations of low-mass hosts, we study satellite galaxy populations as a function of host stellar mass $9.5 < log(M_*/M_odot) < 10.5$ and redshifts $0.1 < z < 0.8$ in the COSMOS survey, making this the first study of satellite systems of low-mass hosts across half the age of the universe. We find that the satellite populations of low-mass host galaxies, which we measure down to satellite masses equivalent to the Fornax dwarf spheroidal satellite of the Milky Way, remain mostly unchanged through time. We observe a weak dependence between host stellar mass and number of satellites per host, which suggests that the stellar masses of the hosts are in the power-law regime of the stellar mass to halo mass relation $(M_*-M_{text{halo}})$ for low-mass galaxies. Finally, we test the constraining power of our measured cumulative luminosity function to calculate the low-mass end slope of the $M_*-M_text{halo}$ relation. These new satellite luminosity function measurements are consistent with ${Lambda}$CDM predictions.
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