3
$\begingroup$

Let $\,n\in\mathbb{N}\,$ and $p$ a prime. Let $P$ be a Sylow $p$-subgroup of $\,S_n\,$. If $p$ does not divide $n$, then $\,P\leq S_{n-1}\,$. Why?

  • 0
    Well, how many times does $p$ divide the order of the group?2012-12-04
  • 0
    The answer is $p^{[n/p]+[n/p^{2}]+...}$. Here [n/p] denotes the largest integer$\leq$n/p2012-12-04
  • 0
    Ok, now compare that to the number of times $p$ divides the order of the smaller group.2012-12-04
  • 0
    It also divides the order of the smaller group.2012-12-04
  • 1
    The way you have written it, it is not true. *Some* of the p-Sylows are subgroups of $S_{n-1}$.2012-12-04
  • 0
    @user I recently got scolded about the addition of the abstract algebra tag when a user tags as group theory. I was directed to a meta post on this topic. So I (who has added abstract algebra to many a questions about groups) was just acting on that reprimand.2013-11-10
  • 0
    @amWhy Well, then something must be wrong with these tags, since as it is described now any group theory (and not only!) question belong to abstract algebra.2013-11-10
  • 0
    @user I agree, wholeheartedly. It was Asaf who called me out on it. I think I'll continue adding it (and allow others to add it too), since I don't believe there is community consensus on the matter, and until the tag description changes (if it changes), I'll simply appeal to your logic, though I won't literally "point" to you. I'll just write out the tag description, as did you.2013-11-10

0 Answers 0