Abelian $\textrm {f.p.f.}$ operator groups of type $(p, p)$

Abstract:

A group A of automorphisms on a group G is said to be fixed-point-free, written f.p.f., if ${C_G}(A) = \{ g \in G|{g^\alpha } = g$ for all $\alpha \in A\} = I$. It has been shown by E. Shult that if A is an abelian f.p.f coprime group of automorphisms of order $n = p_1^{{a_1}} \cdots p_k^{{a_k}}$ on a solvable group G, then the nilpotent length of G is bounded above by $\psi (n) = \sum \nolimits _{i = 1}^{i = k} {{a_i}}$ unless $|G|$ is divisible by primes q such that ${q^s} + 1 = d$ where d divides the exponent e of A. F. Gross has removed the exceptional condition on the prime divisors of $|G|$ when A is cyclic of order ${p^a},p$, p an odd prime. In the case where A is noncyclic of order ${p^2}$, the author has also removed the exceptional condition on the prime divisors of $|G|$.