We study periodic groups of the form 𝐺 left-semidirect-product 𝐹 delimited-⟨⟩ 𝑎 with the conditions subscript 𝐶 𝐹 𝑎 1 and 𝑎 4 . The mapping : 𝑎 → 𝐹 𝐹 defined by the rule → 𝑡 superscript 𝑡 𝑎 superscript 𝑎 1 𝑡 𝑎 is a fixed-point-free (regular) automorphism of the group  𝐹 . In this case, a finite group  𝐹 is solvable and its commutator subgroup is nilpotent (Gorenstein and Herstein, 1961), and a locally finite group  𝐹 is solvable and its second commutator subgroup is contained in the center 𝑍 𝐹 (Kovács, 1961). It is unknown whether a periodic group  𝐹 is always locally finite (Shumyatsky’s Question 12.100 from The Kourovka Notebook  ). We establish the following properties of groups. For 𝜋 𝜋 𝐹 𝜋 subscript 𝐶 𝐹 superscript 𝑎 2 , the group  𝐹 is 𝜋 -closed and the subgroup subscript 𝑂 𝜋 𝐹 is abelian and is contained in 𝑍 superscript 𝑎 2 𝐹 (Theorem 1). A group  𝐹 without infinite elementary abelian superscript 𝑎 2 -admissible subgroups is locally finite (Theorem 2). In a nonlocally finite group  𝐹 , there is a nonlocally finite 𝑎 -admissible subgroup factorizable by two locally finite 𝑎 -admissible subgroups (Theorem 3). For any positive integer  𝑛 divisible by an odd prime, we give examples of nonlocally finite periodic groups with a regular automorphism of order  𝑛 .