find the Number of $ 5-Sylow$ subgroups of the symmetric group $S_5 $?
my solution : $|S_5| $ = $1 \times 2 \times \ 3 \times 4\times 5\times 6 $
$ n_1 = 1 + 5k ||S_5|$
when i take $k = 0, 1$ then it will divide $|S_5|$
Now im confused How to find the 5 sylow subgroups??
Well, since Sylow says it's congruent to 1 mod 5 and divides $24~=~\frac{120}{5}$, and since it can't be $1$ since you can write down more than 4 elements of order 5, simple arithmetic shows that the answer must be 6. (11, 16, 21, etc. do not divide 24.)
What's additionally interesting is that this shows that $S_5$ has a transitive action on the six 5-Sylow subgroups, which gives you a way to embed $S_5$ as a transitive subgroup of $S_6$. $S_5$ also obviously enbeds in $S_6$ as the stabilizer of a point, so it's cool that there is an inequivalent embedding, and in fact this is related to the outer automorphism of $S_6$, which swaps these embeddings.