Periodic Groups Saturated with Finite Frobenius Groups with Complements of Orders Divisible by a Prime Number

A finite Frobenius group in which the order of complements is divisible by a prime number p is called a Φ_p-group. We prove the theorem stating the following. Let G be a periodic group with a finite element a of prime order p > 2 saturated with Φ_p-groups. Then G = F λ H is a Frobenius group with kernel F and complement H. If G contains an involution i commuting with the element a, then H = C_G(i) and F is Abelian, and H = N_G( \(\langle a\rangle \) $⟨a⟩$ ) otherwise.