Research on finite solvable groups with C-closed invariant subgroups has given rise to groups structured as follows. Let p, q 1 , q 2 , . . . , q m be distinct primes, n i be the exponent of p modulo q i , and n be the exponent of p modulo r = m i=1 q i . Then G = P λ x , where P is a group andZ i ; here, Z i and P/Z(P ) are elementary Abelian groups of respective orders p ni and p n , |x| = r, the element x acts irreducibly on P/Z(P ) and on each of the subgroups Z i , and C P (x qi ) = Z i . We state necessary and sufficient conditions for such groups to exist.The study of finite solvable groups with C-closed invariant subgroups has given rise to groups having the following structure. Let p, q 1 , q 2 , . . . , q m be distinct primes, n i be the exponent of a number p modulowhere P is a group and Z(P ) = P = m i=1 Z i ; here, Z i and P/Z(P ) are elementary Abelian groups of orders p ni and p n , respectively, |x| = r, the element x acts irreducibly on P/Z(P ) and on each of the subgroups Z i , and C P (x qi ) = Z i . In the present paper, we argue to state necessary and sufficient conditions for such groups to exist.We introduce the following extra notation: k i = n/n i , r i = r/q i , m i is the exponent of p modulo r i , and l i = n/m i . Then m i = LCM{n j | j = i} and l i = GCD{k j | j = i}.Main THEOREM. A group G structured as above exists if and only if the following hold:(1) n(n−1)(2) all numbers q i are odd;(3) if m > 1, r 0 is a proper divisor of r, and n 0 is the exponent of p modulo r 0 , then n 0 is even and p n0/2 ≡ −1(r 0 );(4) m i = m j for i = j;(5) if m i = n for some i then p n/2 ≡ −1(r); (6) if m > 2 then all l i are odd numbers, greater than 1. Let Z = Z(P ). Note that any degree of x other than 1 acts on P/Z regularly. Indeed, if C P/Z (x k ) = 1 for some number k then the fact that P/Z = [P/Z, x k ] × C P/Z (x k ) and x acts irreducibly on P/Z implies 1É
