Hi all,

This is a question about the groups $H_{n,s}$ introduced by Völklein in his book "Groups as Galois groups", §7.1, and defined as follows: let $N$ be the intersection of all normal subgroups of index $\le n$ in $F_s$, the free group on $s$ generators, and define $H_{n,s} = F_s / N$.

Have these groups been studied? Do they have a name? Is it possible to compute their orders, at least in some cases?

For example for $s=1$, then $H_{n,1}$ is the cyclic group of order $lcm(1, 2, \ldots, n)$.

In Völklein's book, these are introduced primarily to avoid talking about profinie groups (the inverse limits of the $H_{n,s}$, with fixed $s$, is the free profinite group of $s$ generators).

Any information you may have on these will be great appreciated.

Thanks!

Pierre

Hi all,

This is a question about the groups $H_{n,s}$ introduced by Völklein in his book "Groups as Galois groups", §7.1, and defined as follows: let $N$ be the intersection of all normal subgroups of index $\le n$ in $F_s$, the free group on $s$ generators, and define $H_{n,s} = F_s / N$.

Have these groups been studied? Do they have a name? Is it possible to compute their orders, at least in some cases?

For example for $s=1$, then $H_{n,1}$ is the cyclic group of order $lcm(1, 2, \ldots, n)$.

In Völklein's book, these are introduced primarily to avoid talking about profinite groups (the inverse limits of the $H_{n,s}$, with fixed $s$, is the free profinite group of $s$ generators).

Any information you may have on these will be great appreciated.

Thanks!

Pierre