IMPRIMITIVE REGULAR ACTION IN THE RING OF INTEGERS MODULO n

Let n be any positive integer and Z(n) = {0, 1,...,n - 1} be the ring of integers modulo n. Let X-n be the set of all nonzero, nonunits of Z(n), and G(n) the group of all units of Z(n). In this paper, by considering the regular representation pi : G(n) -> Sym(X-n), the following are investigated as follows: (1) G(n) is not fixed-point free; (2) If Fix(g) = {x is an element of X-n : gx = x} not equal empty set for some g is an element of G(n), then Fix(g) is a union of orbits under the regular action of G(n) on X-n; (3) B(subset of o(x(1)) U ... U o(x(l))) is a set of imprimitivity under pi for some orbits o(x(1)),..., o(x(l)) if and only if B = g(1)H(1)x(1)U...Ug(l)H(l)x(l) for some subgroups H-1,.., H-l of G(n) and some elements, g(1),..., g(l) is an element of G(n) satisfying that if (gB) boolean AND B not equal empty set for some g is an element of G, then g is an element of stab(x(i))H-i for each i = 1,...,l.