# Free Group Theory and Free Profinite Group Theory Questions

Dear Experts,

1) Does someone know of any evaluation of the number of $p$-groups of order $p^n$ and rank $k$ ? (In particular, rank $2$? )

2) In the profinite group theory- we take a finitely generated free group $F$ , and denote by $S_f$ the interesection of all the normal subgroups $S \triangleleft F$ such that $F/S$ is a finite group. Does the group $F/S_f$ have a name? (It arises when talking about profinite completions, so I guessed it must have a name)

thanks a lot for any help ! Karrol

• The answer to 2 is the free group. – Benjamin Steinberg Dec 23 '12 at 12:35
• This is residual finiteness of free groups. – Benjamin Steinberg Dec 23 '12 at 12:40
• More generally, for any group G, the intersection of all the normal subgroups of finite index is sometimes called the 'finite residuum' of G. – HJRW Dec 23 '12 at 17:08
• I believe that 1 is totally intractable. – Benjamin Steinberg Dec 23 '12 at 17:38

1-You may consider the function $f(n,p,c,d)$ which is the number of $p$-groups of order $p^n$ of class $c$ generated by at most $d$ generators. This function was studied by Marcus du Sautoy using the theory of zeta functions of groups. You can look at his paper http://www.ams.org/journals/era/1999-05-16/S1079-6762-99-00069-4/S1079-6762-99-00069-4.pdf
Christopher Voll gives an explicit formula for this function when $c=d=2$. You can look at http://arxiv.org/abs/0908.1355