Exploring Stabilizers and Isomorphisms in the Sn Group

[eddyski3]

I have two questions about this group that I think I have an idea about but am unsure of. The first question is how many elements in the Sn group can map 1 to any particular elements, say n-2?
The second question is how do you find the order of the stabilizer of 5 in Sn?

 

[Deveno]

suppose that σ in S_n maps 1→k.then we have n-1 choices for σ(2) (anything but k), n-2 choices for σ(3) (it can't be k or σ(2)), and so on. how many choices will this make in all?

now suppose that σ is in Stab(5). let σ(n) = a (which certainly isn't 5) then (5 n)σ(5 n) takes:

5→n→a→a
n→5→5→n, if a≠n, and

5→n→n→5
n→5→5→n, if a = n.

in either case, we see that (5 n)σ(5 n) is in Stab(n).

thus, the map σ→(5 n)σ(5 n) is an isomorphism of Stab(5) with Stab(n) (it's a bijection because conjugation by any element of a group G is a bijection of G with itself).

but if we have an element of Stab(n), there is a natural isomorphism of this subgroup of S_n with S_n-1, do you see what it is?