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On Thompson's Theorem

✍ Scribed by Lev Kazarin; Yakov Berkovich

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
138 KB
Volume
220
Category
Article
ISSN
0021-8693

No coin nor oath required. For personal study only.

✦ Synopsis

Let p be a prime divisor of the order of a finite group G. Thompson (1970, J. Algebra 14, 129-134) 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 p -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 Ο‡ 1 and Ο‡ 2 of distinct degrees such that p does not divide Ο‡ 1 1 Ο‡ 2 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 p -degree are proved. Some open questions are posed.

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