We classify the irreducible complex characters of prime power degree of the finite quasi-simple groups different from the alternating groups and their covering groups. A well-known example of such characters is given by the Steinberg characters of finite groups of Lie type. It turns out that, apart from sporadic examples coming from numerical coincidences, like Fermat and Mersenne primes, this is the generic case. The proof proceeds via the classification of finite simple groups and makes essential use of Deligne-Lusztig theory.
Introduction.The problem of classification of complex finite linear groups of particular degree attracts attention since the beginning of the century. In the 60s a large contribution to the topic was made by R. Brauer, W. Feit and their successors. After the classification of finite simple groups and huge advances in the character theory for groups of Lie type, this problem becomes much more accessible. The list of primitive finite irreducible linear groups G of prime degree was obtained in [6], see also [12, Corollary on page 420]. In [21] for quasi-simple groups G and primes p dividing |G| there were determined all irreducible characters of degree Ϲ 2 p. In this paper we list quasi-simple irreducible linear groups G (with G/Z(G) not alternating) of degree p k for any k. In other words we list the characters whose degrees are divisible by a single prime.We make heavy use of earlier computations of one of the authors [13] performed for a different purpose. There are hints that the prime divisor structure of degrees of irreducible representations of quasi-simple groups may be useful in various situations and hence deserves further attention. See also [14], [15]. The result is also connected with the problem raised by W. Feit [8] on the existence of p-Steinberg characters for arbitrary finite groups as the degree of such a character is a p-power. For simple groups the problem of Feit has been solved by Tiep [20] who also announced (without a proof) that for simple groups of Lie type G(q) with q > 8 for p being a non-describing characteristic only Weil characters can be of p-power degree. Abdukhalikov [1] observed that if G is a simple group of Lie type in characteristic p then the Steinberg representation is the only non-trivial one of p-power degree unless the Schur multiplier of G is divisible by p.The main result of the present paper is the following:
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