We study periodic groups of the form \(G=F\leftthreetimes\langle a\rangle\) with the conditions \(C_{F}(a)=1\) and \(|a|=4\). The mapping \(a:\,F\to F\) defined by the rule \(t\to t^{a}=a^{-1}ta\) 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 \(\pi=\pi(F)\setminus\pi(C_{F}(a^{2}))\), the group \(F\) is \(\pi\)-closed and the subgroup \(O_{\pi}(F)\) is abelian and is contained in \(Z([a^{2},F])\) (Theorem 1). A group \(F\) without infinite elementary abelian \(a^{2}\)-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 \(n\).

我们在条件\(C_{F}(a)=1\)和\(|a|=4\)下研究形式为\(G=F\leftthreetimes\langle a\rangle\) 的周期群。由规则\(t\to t^{a}=a^{-1}ta\)定义的映射\(a:\,F\to F\)是组 \(F\)。在这种情况下，有限群 \(F\)是可解的，其交换子子群是幂零的（Gorenstein 和 Herstein，1961），一个局部有限群 \(F\)是可解的，它的第二个交换子子群包含在中心\(Z(F)\)（科瓦奇，1961）。未知周期群 \(F\)总是局部有限的（库洛夫卡笔记本中舒米亚茨基的问题 12.100  ）。我们建立以下群的性质。对于\(\pi=\pi(F)\setminus\pi(C_{F}(a^{2}))\)，群 \(F\)是\(\pi\) -closed 和子群\(O_{\pi}(F)\)是阿贝尔量并且包含在\(Z([a^{2},F])\) 中（定理 1）。没有无限初等阿贝尔\(a^{2}\) 可容许子群的群 \(F\)是局部有限的（定理2）。在非局部有限群 \(F\) 中，有一个非局部有限\(a\) -可容许子群可分解为两个局部有限\(a\)- 可容许的子群（定理 3）。对于任何可被奇素数整除的正整数 \(n\)，我们给出了具有\(n\)阶正则自同构的非局部有限周期群的例子 。