Perfect Frobenius complements (Q699733)

\textit{H. Zassenhaus} [in Abh. Math. Semin. Univ. Hamb. 11, 187-220 (1935; Zbl 0011.10302)] and in revised form [in Result. Math. 5, 132-145 (1985; Zbl 0591.20017)], showed that if \(G\) is a perfect complement of a finite Frobenius group then \(G\cong\text{SL}_2(5)\). Both of these proofs involve character theory. \textit{H. Bender} [Group theory, algebra and number theory -- Colloquium in memory of Hans Zassenhaus, Saarbrücken, 1993, 97-143 (1996; Zbl 0872.20001)] gave a further proof of Zassenhaus's theorem based on the elementary theory of exceptional characters. In this note, the author gives two new proofs of Zassenhaus's theorem without using character theory. The first proof is based on standard group theory text book material. The second proof is slightly shorter but uses a couple of simple facts about modular representations.