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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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