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A four group of automorphisms with a small number of fixed points. (English. Russian original) Zbl 0783.20012
Algebra Logic 30, No. 6, 481-489 (1991); translation from Algebra Logika 30, No. 6, 735-746 (1991).
Let \(G\) be a finite soluble group, or more generally a periodic locally soluble group, acted on by a finite soluble group \(A\). Suppose that \(| C_ G(A)|<\infty\). There are many results in the literature relating the structure of \(G\) to the structure of \(A\) and \(| C_ G(A)|\). This paper is a contribution to this area in the case when \(A\) is a four-group. The author begins by mentioning the following conjecture. Let \(| A|\) be the product of \(n\) primes and let \(F_ n(G)\) be the \(n\)th term of the Fitting series of the finite soluble group \(G\). Then \(| G:F_ n(G)|\) is bounded by a number depending only on \(| A|\) and \(| C_ G(A)|\). It is now known that this is false unless one assumes in addition that either \(A\) is nilpotent or that \((| G|,| A|)=1\) [S. D. Bell and the reviewer, Q. J. Math., Oxf. II. Ser. 41, 127-130 (1990; Zbl 0703.20022)]. When \(A\) is nilpotent the situation remains unclear in general, but when \((| G|,| A|)=1\), the conjecture has been proved with \(n\) replaced by \(2n+1\) [I. M. Isaacs and the reviewer, J. Algebra 131, 342-358 (1990; Zbl 0703.20023)].
In this paper the author introduces the derived length of \(G\) as a further parameter in some of his results. The main theorems are as follows. Theorem A. Let \(G\) be a periodic soluble group without involutions, of derived length (solubility length) \(k\), acted on by a four-group \(V\). Assume that \(G=[G,V]\) and that \(| C_ G(V)|\leq m\). Then the derived group \(G'\) of \(G\) has a nilpotent subgroup \(S\), normal in \(VG\), such that \(| G':S|\) is bounded in terms of \(m\) and the class of \(S\) is bounded in terms of \(k\) and \(m\). Theorem B. Let \(G\) be a locally finite group without involutions acted on by a four- group \(V\) such that \(| C_ G(V)|\leq m\). Then \(G\) contains a normal subgroup \(H\) of index bounded in terms of \(m\), satisfying the following: (1) the third term \(\gamma_ 3(H)\) of the lower central series of \(H\) is locally nilpotent, (2) if \(G\) is soluble of derived length \(k\), then \(\gamma_ 3(H)\) is nilpotent of class bounded in terms of \(k\) and \(m\), (3) if \(G\) is hypercentral by soluble, then \(\gamma_ 3(H)\) is hypercentral.

MSC:
20D45 Automorphisms of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20F50 Periodic groups; locally finite groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20E07 Subgroup theorems; subgroup growth
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