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To what extent is there/can there be a description that is uniform in p (for p sufficiently large) of the p-groups of order $p^n$, for each fixed n?

Note 1: I used the word "description" rather than "classification" because I understand that classifying p-groups is notoriously open and difficult. The word "description" is meant to be headed in the general direction of "classification," but somewhat more generous and general.

Note 2: I'm asking more about the possibility of such a uniform-in-$p$ description, and what it could/must/can't look like, rather than just asking about what kind of classification results are currently known or the difficulty of such a classification.

Note 3 (added Feb 10, 2014 2:40pm EST): I changed it from "independent in $p$" to "uniform in $p$" because I think that better captures my intention. I don't mean completely independent of (large) p. As in example (2) and Derek Holt's answer, if there is some description of, say, groups of order $p^{12}$ that depends on, say, the residue class of $p$ mod 20 (a fixed number), and on the number of solutions mod $p$ of a fixed (set of) equation(s) over $\mathbb{Z}$, then I'd be happy with that. The hope is sort of that there should be a finite description, uniform in $p$; that one doesn't need a completely new description for infinitely many values of $p$.

Motivating Examples

1) One of the examples which motivated this question: there is a classification of p-groups of order $p^4$ (going back to Burnside, I believe) for p=2, p=3, and then - crucially for this question - all $p \geq 5$.

2) Another infinite family of partial examples (partial because it only concerns some and not all groups of order $p^n$ for some fixed $n$): isomorphism classes of p-groups of class 2, exponent p, with $G/[G,G]$ of rank $t$ and $[G,G]$ of rank $z$ are in bijective correspondence with orbits of $z$-dimensional spaces of antisymmetric $t \times t$ matrices under change of basis on $G/[G,G]$. I understand that any description of the orbits of the latter action may vary with p (see this related question). But I think I nonetheless want to count this description as "independent of p", since the change-of-basis group and the action can all be defined over $\mathbb{Z}$ and then just taken mod p to get the picture for a given p (if I understand correctly).

3) Added Feb 10, 2:40pm EST: Derek Holt pointed out examples of du Sautoy and Vaughan-Lee suggesting that a classification of groups of order $p^{10}$ should be incredibly difficult. Although I find the difficulty interesting, in this question I'm more interested in the logical/model-theoretic/algebro-geometric/what-have-you possibility of the existence of such a description. For the particular examples in Derek Holt's answer, they give a classification of some groups of order $p^{10}$ that depend only on the value of $p$ mod 12 and on the number of solutions mod $p$ of the pair of integer equations $x^4 + 6x^2 - 3 = y^2 - x^3 + x = 0$. Although this number can vary a lot with $p$, this is still a single, finite description that is uniform in $p$ (and a very interesting example!).

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For each $n$, the groups of order $p^n$ with $p\geq n$ are regular, and so they have type invariants that can help give a generic description. This is what leads Hall to say that "most" $p$-groups are regular (in the sense that for a fixed $n$, only those with $p<n$ may fail to be regular), and this is what underlies the fact that descriptions of groups order $p^n$ tend to break down into $p\geq n$ (the "small class" case) and then special analysis for the cases of $p<n$. –  Arturo Magidin Feb 10 at 17:45
    
Maybe this is the wrong question. There is currently some very fruitful research going on (by Bettina Eick, Max Horn et al.) trying to classify $p$-group not by their order but by their coclass. I'm not familiar with the recent results but if I remember correctly the situation should look something like this: For every fixed $c$ the $p$-groups of coclass $c$ can be organized in an infinite, locally finite tree (forest?) with a certain periodic pattern. There is a correspondence between pro-$p$-group of coclass $c$ and infinite paths in that tree. ... –  Johannes Hahn Feb 10 at 19:07
    
... There are the unavoidable small exception, but for big enough $p$ the tree is fixed I think (maybe I'm wrong). Because the tree is periodic you only have to know a finite part of it to reconstruct the finite $p$-groups of coclass $c$. –  Johannes Hahn Feb 10 at 19:09
    
@JohannesHahn: I am aware of the work on classification by (small) coclass. I am specifically interested in the classification by order, related to algorithms for group isomorphism. The hardest cases seem to be the p-groups groups of class 2, which are of maximal coclass. –  Joshua Grochow Feb 10 at 19:27
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2 Answers

I believe that it is extremely unlikely that there could be such a description. In fact, groups of order $p^n$ for all primes $p$ have been fully classified for $n \le 7$. See here for summary information. Note, in particular, that the number of isomorphism classes increases with $p$ when $n \ge 5$.

The Higman PORC conjecture (polynomial on residue classes) is that, for each $n$, this number is a polynomial function of $p$ and of $p \pmod k$ for a finite collection of values of $k$.

But a family of examples constructed by Vaughan-Lee and du Sautoy of order $p^{10}$ (see here for the full paper, which is $76$ pages long), while not actually disproving the conjecture, suggests that it is very unlikely indeed that it is true.

Just looking quickly at that construction, which depends on the geometry of elliptic curves, will probably help you to understand just how complicated a uniform description of all groups of order $p^{10}$ would be.

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Thanks! The fact that the number of isomorphism classes increases with $p$ for fixed $n \geq 5$ doesn't necessarily rule out a p-independent description, as in Motivating Example 2 (where the number can increase, but there is still a uniform description over $\mathbb{Z}$). The examples of order $p^{10}$ are indeed interesting though - I'll have to look into those further. –  Joshua Grochow Feb 10 at 19:28
    
On looking into it further, I think this is a great example of a description that's uniform in $p$, precisely because it depends on the mod p geometry of a single elliptic curve over $\mathbb{Z}$. This is the kind of behavior I meant to hint at with example (2), but I edited the question with this new example to help clarify. –  Joshua Grochow Feb 10 at 19:52
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I agree that it is still possible to conjecture that, for each fixed $n$, there is some kind of uniform description of this type, possibly involving the solutions mod $p$ of multi-variable polynomial equations, but the available evidence suggests that such descriptions will grow more complicated in unpredictable ways, as $n$ increases. We don't have anything like a complete description for $n=10$, only that of an interesting family of examples. –  Derek Holt Feb 11 at 9:07
    
Ah, I see. I wonder if there is some less refined data that one could get - analogous to the Hilbert polynomial of an invariant ring - that would suggest just how much more complicated such descriptions must grow with $n$... Thanks again! –  Joshua Grochow Feb 11 at 14:52
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For each $n$, the groups of order $p^n$ with $p\geq n$ are regular (a notion defined by Philip Hall). There are many equivalent ways to define regular $p$-groups; one is that if $a$ and $b$ are any elements of the group, then $$(ab)^{p^r} = a^{p^r}b^{p^r}c_1^{p^r}\cdots c_t^{p^r},$$ where $c_i$ are elements of the commutator subgroup of $\langle a,b\rangle$; in essence, a group is regular if the operation of taking $p$th powers interacts "well" with taking commutators.

Hall showed that in a regular $p$-group, one can define "type invariants" which are similar to the invariant factors for finite abelian groups. Though they do not completely determine the groups the way the invariant factors do for abelian groups, they are usually a very good first reduction towards the analysis.

The key observation and why I'm mentioning this in response to your query is that if $p\geq n$, then a group of order $p^n$ is necessarily regular (more generally, if the group is of class $c$ and $p>c$, then the group is regular; in particular, since a group of order $p^n$ is of class at most $n-1$, the observation just made follows); in fact, Hall says in his original paper (Philip Hall, A contribution to the theory of groups of prime power order, Proc. London Math. Soc. (series 2) 36, no. 1 (1934), pp. 29-95 ), if we fix $n$, then "most" groups of order $p^n$ are regular (since only those with $p\lt n$ may fail to be regular).

This leads, classically, to a separation of $p$-groups into those of "small class" (when the class is smaller than $p$), and "the rest".

This also means that when classifying groups of order $p^n$, the analysis usually breaks into two different cases: when the group is regular (which includes all $p\geq n$), and when the group is irregular; the latter case leads to a case-by-case analysis for small primes.

This occurs in the classification of groups of order $p^3$, where we deal with odd primes on the one hand, and then deal with the groups of order $8$ separately. Likewise Burnside's work on group of order $p^4$. The same again occurs in the classification of groups of order $p^6$ (James, Rodney; The groups of order $p^6$ ($p$ an odd prime), Mathematics of Computation 34 no. 150 (1980), pp. 613-637), and $p^7$ (E. O'Brien, M.R. Vaughan-Lee, The groups with order $p^7$ for odd prime $p$, J. Algebra 292 (2005), 243-258); the latter deals with $p\geq 7$ uniformly, and then separately and directly with the groups of order $3^7$ and $5^7$. (The groups of order $2^7$ are dealt with in a completely separate work with different methods, $2$ being the oddest prime of all as usual...)

The latest work uses Lie rings and algebras as a starting point. There are algorithms that are known to produce and check isomorphism types (see the paper by O'Brien and Vaughan-Lee for references). The older classification ($p^6$) was based on Philip Hall's notion of isoclinism as a first, rough classification scheme. In both cases, though, the classification tends to look like a long list of group presentations.

It was hoped, via Higman's PORC conjecture (I see this has just been noted by Derek Holt in his response) that at least a uniform count might exist for all sufficiently large primes; some more recent evidence suggests it might not be quite that simple.

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