## Ways of assigning probability measures on p-groups?

Most of the results/conjectures I'm aware of in asymptotic group theory (involving whether "most" groups satisfy a certain property) assign an equal weight to all groups of the same size. For instance, the conjecture that "most groups are nilpotent" claims that the quotient of the number of nilpotent groups of order at most $n$ to the total number of groups of order $n$ approaches 1. (See this paper for a survey).

But it may also be beneficial to assign different "weights" to different p-groups, similar to the Cohen-Lenstra measure assigned on the set of all abelian p-groups, which weights them by the reciprocal of the size of their automorphism group (see this MathOverflow discussion and this blog post by Terence Tao).

What are the typical or desirable ways of doing this for finite p-groups? In other words, what are the typical measures that we can assign to various sets of finite p-groups, such as:

1. The set of all finite groups of order $p^n$, for fixed n. (Apart from the obvious counting measure).
2. The set of all finite groups of class at most c, for fixed c. (In the case $c = 1$, we have the Cohen-Lenstra measure as mentioned above).
3. The set of all finite groups of given order and satisfying some additional condition, say on its nilpotency class, number of generators, derived length, Frattini length, etc.
-
 If you want to weight using automorphisms, then I suggest looking at Geir Helleloid's papers about automorphism groups of p-groups: arXiv:0711.2816, arXiv:math/0610294 and maybe also arXiv:math/0602039. – Dan Ramras Apr 19 2010 at 17:49