0
$\begingroup$

Let $W:=\prod_{i\in \omega} F_i$ be the (external) unrestricted direct product and $U:=\prod_{i\in \omega}^w F_i$ be the (external) restricted direct product of finite groups $F_i$ such that $\lvert F_{i}\rvert<\lvert F_{i+1}\rvert$ for every $i\in\omega$.

Is it possible to describe the Sylow $p$-subgroups of $W/U$ for a fixed prime $p$ in some ways?

See also Factor group of direct product by restricted direct product.

Improve this question
$\endgroup$
5
  • $\begingroup$ I guess Sylow $p$-subgroup means pro-$p$ subgroup of finite, prime-to-$p$ index? Is it obvious that those exist? $\endgroup$
    – LSpice
    Commented Aug 18, 2021 at 11:32
  • $\begingroup$ By Sylow $p$-subgroup, I mean maximal $p$-subgroups of $W/U$. $\endgroup$
    – IGT
    Commented Aug 18, 2021 at 11:52
  • $\begingroup$ Note that $W/U$ is not in general a torsion group, and might even contain infinite finitely generated $p$-subgroups. The question is likely to have no sensible answer. $\endgroup$
    – YCor
    Commented Aug 18, 2021 at 12:43
  • 1
    $\begingroup$ The assumption that $|F_i|<|F_{i+1}|$ sounds quite artificial/unnecessary in this context, since one can always, otherwise, gather factors together to force it (unless $|F_i|=1$ for large $i$). $\endgroup$
    – YCor
    Commented Aug 18, 2021 at 12:45
  • $\begingroup$ Exponents of my finite groups $F_i$ are strictly increasing but it seems that to make a connection between Sylow $p$-subgroups of $F_i$ s and $W/U$s is not also sensible (maybe with Sylow $p$-subgroups of $W$). $\endgroup$
    – IGT
    Commented Aug 18, 2021 at 13:24

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .