Near-rings of invariants (Q2639144)

A permutation group S acting on a group G written additively, but not necessarily commutative, divides it into equivalence classes, namely the orbits. Starting from a division of G into equivalence classes, we can find an appropriate S. The authors consider the near-ring, denoted by I(S,G) of all mappings from G to G under pointwise addition and composition of maps, which are constant on equivalence classes. Excluding \(S=\{id\}\) this means that I(S,G) is neither zero-symmetric nor has an identity. They obtain a good deal of information about the structure of such near-rings when the number of orbits is finite. Details about the existence or otherwise of various types of ideals or conditions on I(S,G) are given, a characterization of all ideals is given, and so are details of various classes of left ideals (the near-rings are right near-rings). In particular all 0-modular and 2-modular left ideals are characterized, and it is shown that every 0-modular left ideal is 1-modular. The three J radicals are determined and so are the semisimple quotients.