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I have the impression that most people in algebra believe the following statement to be true, but I have no reference for it.

Fix a natural number $n$. Consider for each prime $p$ the set of all groups of order $p^n$. Then is the following true?

  • There is a $p'$ such that for all $p\geq p'$ the number of isomorphism classes of groups of order $p^n$ does only depend on n.
  • We can write down generic presentation for all the groups of order $p^n$ with fixed $n$ and $p\geq p'$. That means that we can give a presentation where each word in the relation subgroup has the same form. It depends only on the chosen prime $p$.
  • Furthermore many group theoretic properties are shared for groups with the same generic presentation (but for different primes). Is it true that these groups have the same nilpotency degree? Is it true that the sizes of the conjugacy classes depend polynomially on $p$? Is it true that the number of cojugacy classes of a certain subgroup also depends polynomially on $p$? What can be said about the ($G$-)poset of subgroups?

I'd like to know what is already known and maybe given a reference.

Thanks in advance.

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I doubt the first bullit point because $n\equiv 1\pmod p$ allows a very "$p$-specific" construction of a semidirect product –  Hagen von Eitzen Jan 30 '13 at 10:18
    
But there are only finitely many such $p$. Now choose $p'$ bigger than any of these $p$. –  Curufin Jan 30 '13 at 10:36
    
Do you mean "depends only on $n$" in the second point? –  Martin Brandenburg Jan 30 '13 at 10:48
    
No. I fix $n$. Generic means for me: I have a presentation where I just replace $p$ with a certain prime. $\langle a\mid a^{p^n}\rangle$ is the generic presentation of a cyclic group with $p^n$ elements. –  Curufin Jan 30 '13 at 10:58
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This is a duplicate of math.stackexchange.com/questions/263876. Your first statement is false for $n=5$. –  Derek Holt Jan 30 '13 at 12:18
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1 Answer

up vote 8 down vote accepted

Your conjectures are sort of true, but the reality is much more complicated than you have phrased it.

Summary

Counting $p$-groups for large $p$ (compared to the nilpotency class) is the same as counting certain finite dimensional (restricted) Lie algebras. Such counts are organized into a tree, with smaller groups being the parents of larger groups. Such counts involve linear algebra and an orbit calculation. The orbit calculation involves counting points in a variety. In all cases known before 2011 or so, the orbit calculations were “PORC”, polynomial on residue classes, so that while no single number, and no single polynomial work, there are finitely many polynomials that work, and the choice of which polynomial is based solely on the residue of $p$ mod some fixed number $n$. Rather than organize $p$-groups by order, it may be better to organize them by coclass: the number of times a child is much larger than the parent. In 2012, a parent group was found so that the orbit calculations involved in a non-PORC way counting the points on a variety. Further investigation found that the conjugacy classes of the group and its descendants were also not PORC.

However, to my mind the calculations for $p^5$ (which are PORC) are extremely similar to these, so to me the essence of the conjecture still holds, but its specific expression is now known to be flawed. I found Vaughan-Lee's recent papers to be a very readable introduction to these ideas, though I learned them from the books mentioned below.

Lazard correspondence

Given a $p$-group $G$, define the subgroups $G_{n+1} = [G,G_n] (G_n)^p$ with $G_1 = G$, so that $G_2 = \Phi(G)$, and $G_n/G_{n+1}$ is an elementary abelian $p$-group centralized by $G$. This called the lower exponent $p$ series. We define a restricted $p$-Lie algebra on $L(G) = \oplus_n G_n/G_{n+1}$ with $[a_i,b_j]_L = v$ and $(a_i)^p_L = w$ where $$v_{\ell} = \begin{cases} [a_i,b_j]_G & \ell = i+j \\ 0 & \text{otherwise} \end{cases} \qquad w_{\ell} = \begin{cases} (a_i)^p_G & \ell = i + 1 \\ 0 & \text{otherwise} \end{cases} $$ In other words, the commutator and $p$th power map are the same in $L(G)$ as in $G$, except that we have to be careful which quotient group everything happens in. If $G_p = 1$, then one can recover the group $G$ from $L(G)$ using the Baker-Campbell-Hausdorff formula for the exponential, so that counting $G$s is the same as counting $L(G)$s.

In Higman's case, this correspondence is fairly clear: $L(G) = G/\Phi(G) \oplus \Phi(G)$ and so every element of $L(G)$ has the form $(\bar g, h)$ and $[ (\bar a, b), (\bar c, d) ] = ( \bar 1, [a,c] )$ and $(\bar a,b)^p = (\bar 1, a^p)$. Any basis of $L(G)$ is a minimal generating set of $G$ (after taking any arbitrary choice of pre-images), and the restricted Lie algebra relations give the relations of the group. Higman showed that if $G_3=1$, then enumerating these Lie algebras was PORC.

$p$-group generation algorithm

To organize the calculation, we view $G/G_n$ as the parent of $G/G_{n+1}$. Given a parent $G/G_n$ that we've already constructed, we try to find all descendants $G/G_{n+1}$. This calculation is described in O'Brien's 1990 article. Again, the gist is just linear algebra and orbit calculations, so one tends to get PORC behavior.

These techniques were used to correct earlier calculations of $p^6$, and to enumerate the groups of order $p^7$. In all cases the answers turn out to be PORC. Each presentation depends on $p$, occasionally requiring elements to be chosen from the orbit space of a variety over some characteristic $p$-field. The properties of each presentation are (more or less by definition of the $p$-group generating algorithm) the same: in particular the nilpotency class and $p$-class is constant on these varieties, indeed the entire structure of $G/G_n$ is constant.

Coclass

Organizing $p$-groups by their order is in many ways a bad idea. Philip Hall suggested using iso-clinism as a better equivalence relation than order, and this method was used in much of the earlier work. However, the coclass proved to be a very nice unifying method. In many cases there is a single (parameterized) expression for an infinite sequence of groups, whose properties (nilpotency class and conjugacy classes) are parameterized in a very simple way. The coclass of $G$ is $n-c$ where $|G|=p^n$ and $c$ is the nilpotency class. A group of coclass 0 is cyclic of order $p$, and groups of coclass 1 are called maximal class. For $p=2$, these are the dihedral, the semidihedral, and quaternion groups; each of these have nice parameterized expressions, and most of the time one has an easy time dealing with the entire family. The coclass conjectures give nice information on the groups in each family using a $p$-adic space group (a single group with a simple presentation whose finite quotients are the mainline groups in that family). The non-mainline groups are topic of current study. du Sautoy's zeta functions, and Eick's computational research group have made significant progress on these groups.

Non-PORC behavior

Recently a group of order $p^9$ whose descendants of order $p^{10}$ are not PORC was discovered by du Sautoy and Vaughan-Lee (2012). In a followup (fairly expository) paper they also show the number of conjugacy classes and the size of the automorphism group are not PORC. In two followup expository papers Vaughan-Lee revisits and simplifies Higman's original PORC calculations.

Bibliography

Books

  • Dixon, J. D.; du Sautoy, M. P. F.; Mann, A.; Segal, D. Analytic pro-p groups. Cambridge University Press, Cambridge, 1999. xviii+368 pp. ISBN: 0-521-65011-9 MR1720368 DOI:10.1017/CBO9780511470882

  • Leedham-Green, C. R.; McKay, S. The structure of groups of prime power order. Oxford University Press, Oxford, 2002. xii+334 pp. ISBN: 0-19-853548-1 MR1918951

  • Holt, Derek F.; Eick, Bettina; O'Brien, Eamonn A. “Handbook of computational group theory.” Chapman & Hall/CRC, Boca Raton, FL, 2005. xvi+514 pp. ISBN: 1-58488-372-3 MR2129747 DOI:10.1201/9781420035216

Articles

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