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.