Matroids on Groups?
Jeremy LeCrone, Nancy Ann Neudauer · College Mathematics Journal · 2014
SummaryBy carefully defining independence, we create two structures on a finite group that satisfy the matroid axioms. Both of these matroids are transversal and graphic, they are duals of each other, and are fundamental transversal matroids. The matroids capture some of the group structure, but two isomorphic matroids may have come from non-isomorphic groups, so we may not be able to recapture the group from the matroid. Our definitions of independent sets cannot be extended in what seems the natural way based on independence of vectors. Finding a definition of independence that satisfies the matroid axioms may not always be possible, though there are always more possibilities on the horizon.