Group actions with inverting centralizers (Q1105035)

Suppose that A, G are finite groups and that A acts as a group of operators on G. The operation is said to have the double centralizer property if \(C_ G(C_ A(H))=H\) for all subgroups H of G and \(C_ A(C_ G(B))=B\) for all subgroups B of A. Theorem 5 states that if this is the case and if \(A=A_ 1\times...\times A_ n\), where \((| A_ i|,| A_ j|)=1\) (i\(\neq j)\), then \(G=G_ 1\times...\times G_ n\), where \((| G_ i|,| G_ j|)=1\) (i\(\neq j)\), \(A_ i\) operates trivially on \(G_ j\) for \(i\neq j\), and \(A_ i\) operates on \(G_ i\) with the double centralizer property. The study of such operations is thus reduced to the case when A cannot be expressed as a non-trivial direct product with factors of coprime order, and Theorem 2 and 3 give necessary and sufficient conditions for the existence of a group G on which A operates with the double centralizer property.