zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] V. V. Belyaev, ”Locally finite groups with Chernikov SylowP-subgroups,” Algebra Logika,20, No. 6, 605–619 (1981). · Zbl 0489.20032
[2] M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of the Theory of Groups [in Russian], 3rd edition, Nauka, Moscow (1982). · Zbl 0508.20001
[3] V. A. Kreknin and A. I. Kostrikin, ”On Lie algebras with regular automorphisms,” Dokl. Akad. Nauk SSSR,149, No. 2, 249–251 (1963). · Zbl 0125.28902
[4] E. I. Khukhro, ”FiniteP-groups admitting an automorphism of orderP-with a small number of fixed points,” Mat. Zametki,38, No. 5, 652–657 (1985).
[5] E. I. Khukhro, ”Nilpotent periodic groups with an almost regular automorphism of prime order,” Algebra Logika,26, No. 4, 502–517 (1987). · Zbl 0663.20039
[6] P. V. Shumyatskii, ”Groups with regular elementary 2-groups of automorphisms,” Algebra Logika,27, No. 6, 715–730 (1988).
[7] S. F. Bauman, ”The Klein group as an automorphism group without fixed points,” Pac. J. Math.,18, No. 1, 9–13 (1966). · Zbl 0144.01702
[8] W. Feit and J. G. Thompson, ”Solvability of groups of odd order,” Pac. J. Math.,13, No. 3, 775–1029 (1963). · Zbl 0124.26402
[9] D. Gorenstein, Finite Groups, Harper and Row, New York (1968).
[10] B. Hartley, ”Periodic locally soluble groups containing an element of prime order with Černikov centralizer,” Quart. J. Math. Oxford,13, No. 131, 309–323 (1982). · Zbl 0495.20011
[11] B. Hartley and Th. Meixner, ”Periodic groups in which the centralizer of an involution has bounded order,” J. Algebra,64, No. 1, 285–291 (1980). · Zbl 0429.20039
[12] B. Hartley and V. Turau, ”Finite soluble groups admitting an automorphism of prime power order with few fixed points,” Math. Proc. Cambridge Philos. Soc.,102, No. 3, 431–441 (1987). · Zbl 0629.20007
[13] G. Higman, ”Groups and rings having automorphisms without nontrivial fixed elements,” J. London Math. Soc.,32, No. 3, 321–334 (1957). · Zbl 0079.03203
[14] O. H. Kegel and B. A. F. Wehrfritz, Locally Finite Groups, North-Holland, Amsterdam (1973). · Zbl 0259.20001
[15] Th. Meixner, Über endliche Gruppen mit Automorphismen, deren Fixpunktgruppen beschränkt sind, Doctoral dissertation, Friedrich-Alexander-Universität, Erlangen-Nürnberg (1979). · Zbl 0436.20015
[16] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 1, Springer-Verlag, Berlin (1972). · Zbl 0243.20032
[17] A. Turull, ”Supersolvable automorphism groups of solvable groups,” Math. Z.,183, No. 1, 47–73 (1983). · Zbl 0505.20016
[18] A. Turull, ”Fitting height of groups and of fixed points,” J. Algebra,86, No. 2, 555–566 (1984). · Zbl 0526.20017
[19] J. Wiegold, ”Groups with boundedly finite classes of conjugate elements,” Proc. R. Soc. London, Ser. A,238, No. 7, 389–401 (1957). · Zbl 0077.03002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.