Finite p-groups with automorphism of a special form

Abstract

Research on finite solvable groups with C-closed invariant subgroups has given rise to groups structured as follows. Let p, q₁, q₂, ..., q_m be distinct primes, n_i be the exponent of p modulo q_i, and n be the exponent of p modulo $r = \prod\limits_{i = 1}^m {q_i }$. Then G = Pλ〈x〉, where P is a group and $Z\left( P \right) = P' = \prod\limits_{i = 1}^m {Z_i }$; Z_i; here, Z_i and P/Z(P) are elementary Abelian groups of respective orders $p^{n_i }$ and pⁿ, ¦x¦ = r, the element x acts irreducibly on P/Z(P) and on each of the subgroups Z_i, and $C_P \left( {x^{q_i } } \right) = Z_i$. We state necessary and sufficient conditions for such groups to exist.