Speaker: John Shareshian, Washington University in St. Louis

Title: Subrack lattices of group racks

Abstract: In joint work with Istvan Heckenberger and Volkmar Welker, we study subrack lattices of group racks. The operation a.b=aba^{-1} makes a group G a rack. The subracks of this rack are those subsets of G that are closed under conjugation. The inclusion partial order makes the set of subracks of G a lattice. Assuming that G is finite, we show that the combinatorial structure of the subrack lattice of G is closely related to the algebraic structure of G. Indeed, if G and H have isomorphic subrack lattices and G is abelian, then H is also abelian. The claim of the previous sentence remains true if we replace "abelian" with "nilpotent", "supersolvable", "solvable" or "simple".