Solubility of finite groups admitting a certain fixed-point-free group of automorphisms
This paper is a further contribution to the conjecture that if the finite group G has a fixed-point free automorphism group A and \((| G|,| A|)=1\), then G is soluble. This was proved by Martineau when A is elementary Abelian and by Rickman when the order of A is the square of a prime. It is proved here when A is an Abelian r-group of type (2,1) and \(C_ G(\Omega_ 1(A))\) is of order a power of an odd prime. This is deduced from Theorem 1.1: if A is soluble, \((| G|,| A|)=1\) and the centre of A contains an elementary Abelian subgroup B of order \(r^ 2\) such that \(C_ G(b)\) is of order a power of an odd prime for every non-identity element b of B, then G is soluble. The above statement follows from this using the results of Martineau and Rickman. After a few preliminary remarks, the bulk of the paper is concerned with the properties of a minimal counterexample to Theorem 1.1.
scientific article; zbMATH DE number 4116793