×

Groups with regular elementary 2-groups of automorphisms. (English. Russian original) Zbl 0739.20017

Algebra Logic 27, No. 6, 447-457 (1988); translation from Algebra Logika 27, No. 6, 715-730 (1988).
Theorem 1. There exists a function \(f(x,y)\) of two natural variables such that any \(k\)-step solvable, periodic group \(G\) admitting a regular elementary group of automorphisms of order \(2^ n\) has an invariant series \(G=H_ 1\supseteq H_ 2\supseteq\cdots\supseteq H_{n+1}=1\), in which the factors are nilpotent and the nilpotent length of \(H_ i/H_{i+1}\) is at most \(f(i,k)\), \(1\leq i\leq n\). Theorem 2. Suppose \(G\) is a periodic group admitting a regular elementary group of automorphisms of order \(2^ n\). If some term of the derived series of \(G\) having a natural subscript is hypercentral, then \(G\) has an invariant series \(G=H_ 1\supseteq H_ 2\supseteq\dots\supseteq H_{n+1}=1\) in which all of the factors are hypercentral.

MSC:

20F50 Periodic groups; locally finite groups
20F14 Derived series, central series, and generalizations for groups
20F28 Automorphism groups of groups
20F16 Solvable groups, supersolvable groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] V. A. Kreknin and A. I. Kostrikin, ”On Lie algebras with a regular automorphism,” Dokl. Akad. Nauk SSSR,149, No. 2, 249–251 (1963). · Zbl 0125.28902
[2] 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
[3] T. R. Berger, ”Nilpotent fixed-point-free automorphism groups of solvable groups,” Math. Z.,131, No. 4, 305–312 (1973). · Zbl 0257.20018 · doi:10.1007/BF01174905
[4] E. Shult, ”On groups admitting fixed-point-free Abelian operator groups,” Illinois J. Math.,9, No. 4, 701–720 (1965). · Zbl 0136.28504
[5] A. Turull, ”Supersolvable automorphism groups of solvable groups,” Math. Z.,183, No. 1, 47–73 (1983). · Zbl 0505.20016 · doi:10.1007/BF01187215
[6] A. Turull, ”Fitting height of groups and of fixed points,” J. Algebra,86, No. 2, 555–566 (1984). · Zbl 0526.20017 · doi:10.1016/0021-8693(84)90048-6
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.