1
$\begingroup$

Problem statement :

Let $G=S_5$ (it actually reads $|G|=S_5$ but this must be a misprint obviously). Using Sylow , how many 5-Sylow subgroups has G got? Is this consistent with the number of 5 cycles in G?

My proof:

we have $|G|=|S_5|=120=5\cdot 24$.

By Sylow's first theorem we now know that the order of the $5$-Sylow subgroups is 5.

By Sylow's 3rd theorem we know that the number of $5$-Sylow n_5=6.$

if $n_5=1$ then we also know that $|N_G(P)|=|G|$ but then this would imply that the normaliser contains every element in $G$ But of course $S_n$ is non-abelian for $n \geq 3$ so this can't be true.

We conclude that that $n_5=6$. So there are 6 $5$-Sylow -subgroups each with order $5$. These are distinct because the intersection is itself a subgroup and by Lagrange's theorem the intersection is either $1$ or $5$ and so must be $1$. so they intersect only at the identity. so any $5$-Sylow subgroup has order $5$ and contains the identity and $4$ $5$-cycles. Overall this means there are $(6\times 4=)\:24$ $5$-cycles, the same as would be expected in $S_5.$

$\endgroup$
  • $\begingroup$ "... be expected in $S_4$"??? $\endgroup$ – user10354138 Nov 6 '18 at 21:30
  • $\begingroup$ sorry that was a typo $\endgroup$ – excalibirr Nov 6 '18 at 21:31
  • $\begingroup$ $S_5$ nonabelian doesn't mean it cannot contain a proper normal subgroup. $\endgroup$ – user10354138 Nov 6 '18 at 21:33
  • $\begingroup$ yes I had started to grow concerned over that bit being wrong after I'd posted it. is all the rest correct though ? $\endgroup$ – excalibirr Nov 6 '18 at 21:35
  • 1
    $\begingroup$ The rest is OK (athough I wouldn't call calculating the order of a Sylow $p$ the content of the first Sylow theorem). To conclude $n_5=6$, look at the subgroup $A_5$ which is simple. $\endgroup$ – user10354138 Nov 6 '18 at 21:38

protected by Community Nov 8 '18 at 16:22

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?

Browse other questions tagged or ask your own question.