@Shahrooz. After some thought I did not arrive to the conclusion that how you can derive the uniqueness of the irreducible character of degree $n-1$ From these two papers which are about nonzero characteristic. The symmetric group of degree $n$ has two irreducible character of degree $n-1$ corresponding to the partitions $n-1,1$ and $2, 1^{n-2}$. I think one must show that these two irreducible characters give only one character for the alternating. Actually I need an explicite reference as Derek Holt mentioned above it is widely known since it is quoted in the Wikipedia.

Many thanks. Is it possible to find such a group $G$ of any derived length $d>2$ such that $G$ is indecomposable? By an indecomposable group, I mean a group which cannot be written as a direct product of two nontrivial subgroups.

@Marco: not yet!

I do not think the problem is a homework question as I have not seen it in any famous undergraduate algebra textbook that I am familiar with.