In 1998, the second author of this paper raised the problem of classifying the
irreducible characters of Sn of prime power degree. Zalesskii proposed the analogous
problem for quasi-simple groups, and he has, in joint work with Malle, made
substantial progress on this latter problem. With the exception of the alternating
groups and their double covers, their work provides a complete solution. In this
article we first classify all the irreducible characters of Sn of prime power degree
(Theorem 2.4), and then we deduce the corresponding classification for the alternating
groups (Theorem 5.1), thus providing the answer for one of the two remaining
families in Zalesskii's problem. This classification has another application in group
theory. With it, we are able to answer, for alternating groups, a question of Huppert:
which simple groups G have the property that there is a prime p for which G has
an irreducible character of p-power degree > 1 and all of the irreducible characters
of G have degrees that are relatively prime to p or are powers of p?
The case of the double covers of the symmetric and alternating groups will be dealt
with in a forthcoming paper; in particular, this completes the answer to Zalesskii's
problem.
The paper is organized as follows. In Section 2, some results on hook lengths in
partitions are proved. These results lead to an algorithm which allows us to show
that every irreducible representation of Sn with prime power degree is labelled by
a partition having a large hook. In Section 3, we obtain a new result concerning
the prime factors of consecutive integers (Theorem 3.4). In Section 4 we prove
Theorem 2.4, the main result. To do so, we combine the algorithm above with
Theorem 3.4 and work of Rasala on minimal degrees. This implies Theorem 2.4 for
large n. To complete the proof, we check that the algorithm terminates appropriately
for small n (that is, those n [les ] 9.25 · 108)
with the aid of a computer. In the last section we derive the classification of irreducible characters of
An of prime power
degree, and we solve Huppert's question for alternating groups.