The problem of classifying pairs consisting of a finite Abelian group and a subgroup leads to the study of rooted trees whose nodes are decorated with natural numbers that strictly increase as you go towards the root. The lattices of trees that correspond to indecomposable pairs that are bounded by pn were generated by computer up to n = 6. As a result, an unexpected, almost complete, duality was discovered in these lattices. The structure of a finite Abelian group is completely described by

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... ple numerical invariants: each such group can be written as a direct sum of cyclic groups of orders nt, n2 ..... nm such that n~ > 1 and ni+ t is a multiple of ni for i = 1 ..... m -1. The numbers ni are unique even though the group can be written as a direct sum of cyclics in many different ways. A much more complicated problem is to classify the subgroups of a finite Abelian group, paying attention not only to the structure of the subgroup, but how it sits inside the group. As is the case for finite Abelian groups themselves, this problem can easily be reduced to the study of p-groups --groups in which each element has order a power of p, for some fixed prime p. Henceforth, the word "group" will refer to a finite p-group. Every rooted tree can be thought of as a presentation of a p-group, via generators and relations, as follows: the generators are the nodes of the tree, we set pr = 0 for the root r, and we set px = y