A new subgroup, the endocenter, is defined. The endocenter is a "functorial center". The endocenter also facilitates identification of groups associated with quasigroup modules. We use the endocenter to investigate classes of quasigroups whose combinatorial multiplication group is universal, and classes of quasigroups whose combinatorial multiplication group is not universal;There is a strong connection between groups with triality and Moufang loops. We give a partial classification of those Moufang loops whose combinatorial multiplication group is with triality. We completely characterize all groups with triality associated with cyclic groups. We also identify some universal multiplication groups of Moufang loops and determine their triality status;Unfortunately, the class of groups with triality is not a variety. In an attempt to overcome this apparent deficiency, we axiomatize the variety of "triality groups", and initiate an algebraic investigation of this (and related) varieties;There are strong geometric connections between Moufang loops and groups with triality. We investigate some of these connections;A new class of groups associated with Moufang loops, but more general than the class of groups with triality, is defined. This is the class of groups with biality. We investigate groups with biality and obtain abstract characterizations of multiplication groups of various classes of inverse property loops.