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I would like to understand the concept of classification in the context finite groups. For finite abelian groups and finite simple groups, it's clear to me what is meant by classification. However, these are the cases where a "perfect classification" turns out to work fine.

Because I have a pretty good understanding of nilpotent groups, I asked myself how the generalization of the classification of finite abelian groups to finite nilpotent groups would look like. In a certain sense, the statement "Every finite nilpotent group is the direct product of p-groups" contains the classification of finite abelian groups as a special case. So at least the classification of finite nilpotent groups is now reduced to the classification of finite (indecomposable) p-groups. But even so the structure of p-groups is "relatively simple" and well understood, trying to enumerate the isomorphism classes is challenging and doesn't seem to add much value.

Would an acceptable solution for the classification problem of finite nilpotent groups necessarily include the possibility for an enumeration of the isomorphism classes? Or are "more practical" solutions also acceptable? For example, would it be sufficient to give both an "efficient" algorithm for constructing at least one representative of each isomorphism class of a given order, and an "efficient" algorithm for deciding whether two representatives corresponds to the same isomorphism class? What would be the role/importance (with respect to the classification problem) of characteristic numbers that only depend on the isomorphism class?

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There are several partial classifications of nilpotent groups already "out there". There's Phillip Hall's work on the regular $p$-groups, which includes the small class case (when the class is smaller than the prime; this is a direct generalization of the abelian case, and looks very similar to it). There's the classification of $p$-groups of maximal class. More recently, there's the work on classification by coclass, and the proof of the Coclass Conjectures. But even "nice" classes are notoriously difficult (e.g., class 2 and prime exponent). –  Arturo Magidin Jan 3 '12 at 21:23
    
It's an open problem if there's a polynomial time algorithm to determine whether or not two p-groups are isomorphic see here. –  JSchlather Jan 3 '12 at 21:24
    
G.Higman and C. Sims gave asymptotic estimates for the number of isomorphism types of $p$-groups of a given order. The numbers are very large. –  Geoff Robinson Jan 3 '12 at 21:38
    
I assume these papers are well known: ams.org/mathscinet-getitem?mr=1283739 and ams.org/mathscinet-getitem?mr=2166803 –  Jack Schmidt Jan 3 '12 at 22:16
    
@JacobSchlather Trying to find a polynomial time algorithm (polynomial in the group order) to determine whether or not two p-groups are isomorphic is more practical and realistic than trying to enumerate the isomorphism classes. I wonder whether the algorithm in the paper linked by JackSchmidt is actually polynomial time or not. I just checked that my local university library allows me to access the paper, but I'm not sure whether/when I will really take the effort to obtain and read it. –  Thomas Klimpel Jan 3 '12 at 23:44

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