Finite p-groups with automorphism of a special form

Research on finite solvable groups with C-closed invariant subgroups has given rise to groups structured as follows. Let p, q1, q2, ..., qm be distinct primes, ni be the exponent of p modulo qi, and n be the exponent of p modulo r = π i = 1m qi . Then G = Pλã(x), where P is a group and Z(P) = P′ = πi = 1mZ i ; Zi; here, Zi and P/Z(P) are elementary Abelian groups of respective orders pni and pn, |x| = r, the element x acts irreducibly on P/Z(P) and on each of the subgroups Z i, and CP(xqi) = Zi. We state necessary and sufficient conditions for such groups to exist. © Springer Science+Business Media, Inc. 2006.