Abstract: The structure of finite abelian groups is well-known and easy to describe: they are direct sums of indecomposables, and the indecomposables are cyclic groups of order $p^n$, where $p$ is a prime number. It follows that the isomorphism classes of finite abelian $p$-groups correspond bijectively to the partitions. If we look at subgroups of such groups, they can be described again in this way. But we want to keep track of the embedding, thus we want to find normal forms for the pairs $(A_0,A_1)$ where $A_0$ is a finite abelian $p$-group and $A_1$ a subgroup of $A_0$. This problem was posed in 1934 by Birkhoff: he showed that the difficulties increase with the growth of the exponent of $A_0$ and he pointed out that the first real problem arises in case the exponent of $A_0$ is equal to 6.
There is a related problem: to classify invariant subspaces of a nilpotent operator on a vector space. The general situation to be discussed is the following: Let $\Lambda$ be a commutative uniserial ring of length $n$ (for example $\Bbb Z/p^n$ or $k[T]/T^n$, where $k$ is a field). Consider the category $\Cal S(\Lambda)$ of pairs $(A_0,A_1)$, where $A_0$ is a $\Lambda$-module and $A_1$ a submodule. In case $n \le 5$, the category $\Cal S(\Lambda)$ contains only finitely many indecomposable objects, namely 5, 10, 20, or 50, in case $n = 2,3,4,$ or $5$, respectively. It is of interest that these cases are related to the Dynkin diagrams $A_2, D_4, E_6$ and $E_8$, but also that the classification is independent of $\Lambda$.
The case $n=6$ is of special interest: for $\Lambda = k[T]/T^6$ a full classication of the indecomposable objects can be given, using two parameters (in $\Bbb Q$ and in $\Bbb N$) which are purely combinatorially, and in addition one which takes values in the projective line over the field $k$. One may wonder whether there is a similar classification for $\Lambda = \Bbb Z/p^6.$ For any $n$, the categories $\Cal S(\Lambda)$ are (non-abelian) Frobenius categories and the corresponding stable categories seem to be of special interest. The distinction between the cases $n<6$, $n=6$ and $n>6$ (finite, tame and wild representation type) may be formulated as elliptic, parabolic or hyperbolic behaviour.
The results presented are joint work with Markus Schmidmeier (Florida Atlantic University).