A Note on Frobenius Groups

✍ Scribed by Paul Flavell

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 87 KB
• Volume: 228
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2000.8269

✦ Synopsis

Two celebrated applications of the character theory of finite groups are Burnside's p ␣ q ␤ -Theorem and the theorem of Frobenius on the groups that bear his name. The elegant proofs of these theorems were obtained at the beginning of this century. It was then a challenge to find character-free proofs of these results. This was achieved for the p ␣ q ␤ -Theorem by w x w x w x Bender 1 , Goldschmidt 3 , and Matsuyama 8 in the early 1970s. Their proofs used ideas developed by Feit and Thompson in their proof of the Odd Order Theorem. There is no known character-free proof of Frobenius' Theorem.

w x Recently, Corradi and Horvath 2 have obtained some partial results in ´this direction. The purpose of this note is to prove some stronger results. We hope that this will stimulate more interest in this problem.

Throughout the remainder of this note, we let G be a finite Frobenius group. That is, G contains a subgroup H such that 1 / H / G and H l H g s 1 for all g g G y H.

A subgroup with these properties is called a Frobenius complement of G.

The Frobenius kernel of G, with respect to H, is defined by

In the language of permutation groups, this corresponds to having a transitive permutation group in which any two-point stabilizer is trivial.