Fixed-point-free automorphism groups from rings. (Q2569315)

A nonidentity automorphism \(\varphi\) of a group \((N,+)\) is called fixed-point-free if \(\varphi(x)=x\) if and only if \(x=0\). An automorphism group \(\Phi\) of \(N\) is called fixed-point-free if every nonidentity automorphism in \(\Phi\) is fixed-point-free. In this case, \((N,\Phi)\) is referred to as a Ferrero pair. A Ferrero pair is called indecomposable if there are no \(\Phi\)-invariant subgroups \(N_1\) and \(N_2\) such that \(N=N_1\oplus N_2\). Let \((R,+,*)\) be a ring with \(1\) and let \(R^*\) denote the multiplicative group of units of \(R\). Then \(R^*\) acts as an automorphism group on \((R,+)\) by multiplication on the left. If \(\Phi\) is a subgroup of \(R^*\) that acts fixed-point-freely on \((R,+)\), then \((R,\Phi)\) is called a ring-generated Ferrero pair. In this article, it is shown that if \((N,\Phi)\) is a finite indecomposable Ferrero pair with \(N\) Abelian, then \((N,\Phi)\) can be embedded into the ring-generated Ferrero pair \((R,\Phi)\) where \(R\) is the ring generated by \(\Phi\) in the ring of endomorphisms \(\text{End}(N)\) of \(N\). Since every finite Ferrero pair \((M,\Theta)\) with Abelian \(M\) can be written as a sum \((M_1\oplus M_2\oplus\cdots\oplus M_k,\Theta)\) with each \((M_i,\Theta)\) indecomposable, this means that \((M,\Theta)\) can be embedded into a sum of ring-generated Ferrero pairs.