Half-transitive group actions in a compact ring
Below R will be a compact ring with identity, G the group of units of R and X denotes the set of all nonzero non-units of R. The authors continue the study of the properties of the regular and conjugate actions of G on X [see J. Pure Appl. Algebra 51, 231-239 (1988; Zbl 0649.16036) and ibid. 54, 167-179 (1988; Zbl 0662.16025)]. By definition G is half-transitive by regular action on X if G is transitive on X or \(| Gx| =| Gy|\) for all \(x,y\in X\) and for each \(x\in X\) Gx is a finite set having more than one element. For a compact ring R the group G is half-transitive on X iff R is isomorphic to one of the following rings: 1) a finite field; 2) \(F\times F\) where F is a finite field and \(| F| >2\); 3) the ring of \(2\times 2\) matrices over a finite field; 4) a finite local ring S such that \(J^ 2=\{0\}\) and J is a left one-dimensional vector space over S/J, where J is the Jacobson radical of S; 5) a local ring S such that S/J is a finite field, \(| S/J| >2\) and \(J^ 2=\{0\}.\) If G is half-transitive on X by conjugate action and \(X\neq \emptyset\), then there exists an \(x\in X\) with \(| O(x)| \neq 2\), where O(x) is the orbit of x. If, in addition, G is not transitive on X, then the characteristic of R is a prime number p. The ring R is local iff \((p,| O(x)|)=1\) for some \(x\in X\).
scientific article; zbMATH DE number 4132393