Let p be a prime divisor of the order of a finite group G. Thompson (1970, J. Algebra14, 129ā134) has proved the following remarkable result: a finite group G is p-nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p (in fact, in that case G is solvable). In this note, we prove that a group G, having only one nonlinear irreducible character of pā²-degree is a cyclic extension of Thompson's group. This result is a consequence of the following theorem: A nonabelian simple group possesses two nonlinear irreducible characters Ļ1 and Ļ2 of distinct degrees such that p does not divide Ļ1(1)Ļ2(1) (here p is arbitrary but fixed). Our proof depends on the classification of finite simple groups. Some properties of solvable groups possessing exactly two nonlinear irreducible characters of pā²-degree are proved. Some open questions are posed.