A noncoprime Shult type theorem
Let G be a finite p-solvable group and let V be a faithful KG-module, K a field. Assume that G contains an element x of order \(p^ n\), p a prime, acting fixed point freely on V. Conditions are given assuring that \(x^{p^{n-1}}\in O_{p'p}(G)\). If char(K) does not divide the order of \(O_{p'}(G)\) we state results assuring that \(x^{p^{n-1}}\in O_ p(G)\). These results extend Shult's theorem. A new proof of Thompson's conjecture asserting that a finite p-solvable (resp. solvable) group G admitting an automorphism \(\alpha\) of order \(p^ n\) acting fixed point freely on every \(\alpha\)-invariant p'-section of G has its p-length (resp. Fitting length) bounded by some functions of n is given for p odd. \textit{B. Hartley} and \textit{A. Rae} [Bull. Lond. Math. Soc. 5, 197-198 (1973; Zbl 0273.20015)] showed that \(l_ p(G)\leq 2n\) and \(f(G)\leq 2n^ 2+3n\). Here the best possible bounds \(l_ p(G)\leq n+1\) and \(f(G)\leq 2n+1\) are obtained.
scientific article; zbMATH DE number 3989624