Let \(A\) be a group of automorphisms of the group \(G\) and let \(C(A)\) be the centralizer near-ring determined by \(A\) and \(G\). If \(G\) is finite, C. J. Maxson and K. C. Smith [Commun. Algebra 8, 211-230 (1980; Zbl 0425.16028)] have shown that \(C(A)\) is simple if and only if all stabilizers \(St(g)\), \(g\in G\), are maximal and conjugate. Two theorems on the general case are proved here. First if \(A\) is nilpotent and \(C(A)\) is simple, then all stabilizers are maximal and conjugate. Second, a counterexample is constructed in which \(C(A)\) is simple, there is no maximal stabilizer and not all stabilizers are conjugate. In this example \(G\) is a countable dimensional vector space over a countable field and \(A\) is soluble of class 2.
For the entire collection see [Zbl 0856.00015].