Finite groups with a~Frobenius subgroup

A. V. Romanovskii

Sbornik. Mathematics, Tome 36 (1980) no. 4, pp. 577-601

Résumé

Suppose $M$ denotes a $CC$-subgroup of order $m$ of a group $G$ which is different from its normalizer in $G$. A criterion for the simplicity of a group is obtained which includes the theorems of Feit and Ito on Zassenhaus groups of even degree and which is used to prove the following Theorem. If $|G:N(M)|=m+1$ and the order of the centralizer of each nonidentity element of $N(M)$ in $G$ is odd, then $G\simeq PSL(2,m)$. It is proved that if $M$ has a complement $B$ in $G$ and if $|M|-1$ does not divide $|B|$, then $N(M)$ has a nilpotent invariant complement in $G$, and if $M$ is complemented by a Frobenius subgroup in the simple group $G$, then $G\simeq PSL(2,2^n)$, $n>1$. Related to the results of Brauer, Leonard, and Sibley on finite linear groups is the following Theorem. {\it If the degree of each irreducible constituent of some faithful complex character $\varphi$ of $G$ is less than $(m-1)/2$, then either $M\lhd G$ or $G\simeq Sz(2^{2n+1})$, $n\geqslant1$.} Other results connected with the above theorems are also obtained. Bibliography: 24 titles.