Richness of the subgroup structure of p-groups Given a prime $p$ and $n \in \mathbb{N}$, let $f_p(n)$ be the smallest
number such that there is a group of order $p^{f_p(n)}$ which all groups of
order $p^n$ embed into. What is the asymptotic growth of $f_p(n)$ when
$n$ tends to infinity?
The question asks in a certain sense for how dense $p$-groups can be "packed together"
as subgroups of a larger group.
Let's give an example for illustration: By the bound by Francois Brunault, all groups of
order $2^{20}$ embed into a group of order $2^{2^{20}-1}$, which is a number with
315653 decimal digits. On the other hand, by Nick Gill's bound, they do not embed
into a group of order $2^{66}$, which is a 20-digit number.
Can these bounds be refined?
Added on Feb 21, 2013: Even if finding precise asymptotics for $f_p(n)$ turns out to be
delicate, isn't it at least possible to decide whether $f_p(n)$ grows polynomially or
exponentially, or whether its growth rate lies somewhere in between?
Or alternatively, are there reasons to believe that this is a difficult problem?
Added on Dec 4, 2013: The question whether it is true that $f_p(n)$ grows faster than polynomially but slower than exponentially when $n$ tends to infinity will appear as Problem 18.51 in:
Kourovka Notebook: Unsolved Problems in Group Theory. Editors V. D.
Mazurov, E. I. Khukhro. 18th Edition, Novosibirsk 2014.
 A: The comment by Frieder Ladisch suggests to me that considering exponents may be relevant. Suppose that we generalize Stefan Kohl's function
$f_p(n)$ as follows:
Definition: Fix a prime $p$ and an exponent $e$. Let $F(p,e,n)$ be the smallest integer such that there is a group of order $p^{F(p,e,n)}$ which contains
isomorphic copies of every group of order $p^n$ and exponent $p^e$.
Lemma: Then $\max_{1\leq e\leq n} F(p,e,n)\leq f_p(n)\leq \sum_{e=1}^n F(p,e,n)$ for all $n\geq1$ and all primes $p$.
Proof: The upper bound is obtained by considering direct products, and the lower bound is easy.
Clearly $F(p,n,n)=n$ and $F(2,1,n)=n$ as a group order $p^n$ and exponent $p^e$ is cyclic if $n=e$, and is elementary abelian if $p=2$ and $e=1$. A very wild guess is that the asymptotic size of $f_p(n)$ as $n\to\infty$ is governed by $F(p,1,n)$ for $p>2$, and $F(2,2,n)$ for $p=2$.
It is unclear to me how helpful wreath products are. Suppose that
$G(p,e,n)$ is a $p$-group that contains isomorphic copies of every group of order
$p^n$ and exponent $p^e$. I claim (without proof) that the $p$-group $G(p,e_2,n_2)\;{\rm wr}\;G(p,e_1,n_1)$ contains isomorphic copies of every group of order
$p^{n_1+n_2}$ and exponent $p^{e_1+e_2}$. This gives the upper-bound $$F(p,e_1+e_2,n_1+n_2)\leq F(p,e_2,n_2)p^{F(p,\,e_1,\,n_1)}+F(p,e_1,n_1).$$
In terms of the previous discussion, a Sylow $p$-subgroup of $S_{p^e}$ or ${\rm GL}(e+1,p)$ has exponent $e$.
A: Each finite $p$-group $H$ may be embedded in the Frattini subgroup of an appropriate $p$-group $G$ (for example, $G=H \wr C_p$; here $d(G)\le d(H)+1$. Then for this $G$, the Frattini subgroup $\Phi(G)$ contains a subgroup which is isomorphic to $H$.)
Problem. Is it true that one can choose $G$ so that $d(G)=2$?
Yakov
