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

  • 2
    $\begingroup$ The answer to 2 is the free group. $\endgroup$ Dec 23, 2012 at 12:35
  • 1
    $\begingroup$ This is residual finiteness of free groups. $\endgroup$ Dec 23, 2012 at 12:40
  • $\begingroup$ More generally, for any group G, the intersection of all the normal subgroups of finite index is sometimes called the 'finite residuum' of G. $\endgroup$
    – HJRW
    Dec 23, 2012 at 17:08
  • $\begingroup$ I believe that 1 is totally intractable. $\endgroup$ Dec 23, 2012 at 17:38

1 Answer 1


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



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.