Rozhkov, A. V. On the theory of Alëshin type groups. (Russian) Zbl 0614.20019 Mat. Zametki 40, No. 5, 572-589 (1986). Let A be a sequence of sets \(A_ 0,A_ 1,..\). each of which contains at least two elements, Cort A be the (partially) ordered set of all finite sequences of the form \((a_ 0,a_ 1,...,a_ n)\), \(a_ i\in A_ i\), with the partial ordering \(\leq\) where \(\mu\leq \nu\) means the sequence \(\mu\) is an initial segment of the sequence \(\nu\). The support of a transformation f of Cort A is, by definition, the set supp f\(=\{n\in {\mathbb{N}}|\) \((\nu f)_ n\neq \nu_ n\) for some \(\nu\in Cort A\}\) where \(\nu_ n\) denotes the \((n+1)th\) member of \(\nu\). An automorphism of the ordered set Cort A is said to be a mutation if it changes at most one member of any sequence. A mutation f is said to be a root mutation if it acts only on the first members of sequences as some permutation \(\pi_ 0=\pi_ 0(f)\in Sym A_ 0\). Let \(\gamma\) be a fixed infinite sequence \(\gamma =(a_ 0,a_ 1,...)\), \(a_ i\in A_ i\). A mutation g is said to be a longitudinal mutation (along \(\gamma)\) if supp f is infinite, \(\gamma f=\gamma\), and, for every sequence \(\eta\neq \gamma\) with the smallest number n such that \(\eta_ n\neq \gamma_ n\), g acts on \(\eta_{n+1}\) as some permutation \(\pi_{n+1}=\pi_{n+1}(g,\eta_ n)\in Sym A_{n+1}\) depending only on g, n, \(\eta_ n\) (a priori, it is admitted that \(\pi_{n+1}\) may be dependent on \(\gamma\) also but, by Lemma 1, \(\gamma\) is fully defined by g) and fixes all other members of \(\eta\). Let C be a subgroup of root mutations, and D be a subset of longitudinal ones of Cort A. If the groups \[ \Pi_ 0=\Pi_ 0(C)=gp(\pi_ 0(f)| \quad f\in C), \]\[ \Pi_ n=\Pi_ n(D)=gp(\pi_ n(g,a)| \quad g\in D,\quad a\in A_{n-1}),\quad n=1,2,..., \] are all transitive then the group \(H=gp(C,D)\) is said to be an Alëshin type group or, briefly, an AT-group over the sequence A with the accompanying permutation groups \(\Pi_ 0,\Pi_ 1,..\). This notion is a wide generalization of that from S. V. Alëshin’s article [ibid. 11, No.3, 319-328 (1972; Zbl 0246.20024)]. The author establishes first some general properties of AT-groups including relations in them and some properties of arrangement of these groups in the full automorphism group Aut Cort A. A criterion for periodicity of AT-groups of certain important type (so-called \(AT_{\omega}\)-groups) is given (Theorem 1), and some sufficient conditions for periodicity of groups of a more general type are indicated (Theorem 2). A 2-generated periodic AT-group is also constructed which contains elements of arbitrary finite order. Reviewer: Yu.I.Merzlyakov Cited in 8 ReviewsCited in 10 Documents MSC: 20F05 Generators, relations, and presentations of groups 20F50 Periodic groups; locally finite groups 20B27 Infinite automorphism groups 20B30 Symmetric groups Keywords:finite sequences; longitudinal mutation; root mutations; Alëshin type group; permutation groups; AT-groups; periodicity; 2-generated periodic AT-group Citations:Zbl 0246.20024 PDFBibTeX XMLCite \textit{A. V. Rozhkov}, Mat. Zametki 40, No. 5, 572--589 (1986; Zbl 0614.20019)