  A "biased expansion" of a graph is a kind of branched covering graph with
additional structure related to combinatorial homotopy of circles. Some but not
all biased expansions are constructed from groups ("group expansions"); these
include all biased expansions of complete graphs (assuming order at least
four), which correspond to Dowling's lattices of a group and encode an iterated
group operation. A biased expansion of a circle with chords encodes a multary
(polyadic, n-ary) quasigroup, the chords corresponding to factorizations, i.e.,
associative structure. We show that any biased expansion of a 3-connected graph
(of order at least four) is a group expansion, and that all 2-connected biased
expansions are constructed by expanded edge amalgamation from group expansions
and irreducible multary quasigroups. If a 2-connected biased expansion covers
every base edge at most three times, or if every four-node minor is a group
expansion, then the whole biased expansion is a group expansion. In particular,
if a multary quasigroup has a factorization graph that is 3-connected, if it
has order 3, or if every residual ternary quasigroup is an iterated group
isotope, it is isotopic to an iterated group. We mention applications to
generalizing Dowling geometries and to transversal designs of high strength.
