Let G be a finite abelian p-group of type lambda. It is well-known that the lattice L(p) of subgroups of G is the order-theoretic p-analogue of the chain product [0, lambda]. However, any surjection &phis; : L(p) → [0, lambda] with order analogue properties does not respect group automorphisms. We are interested in L, the quotient lattice of L(p) under the action of a Sylow p-subgroup of the automorphism group of G. This quotient lattice is particularly interesting since it respects group automorphisms, has the property that the size of an orbit of the action is a power of p, and is closely related to the product of chains [0, lambda]. We will discuss combinatorial properties of L as well as interesting properties of quotients of L(p) under the actions of lattice automorphisms and lattice automorphisms induced by group automorphisms that arise in the course of studying L.
展开▼
