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A solvable group admitting a regular splitting automorphism of prime order is nilpotent. (English. Russian original) Zbl 0426.20027
Algebra Logic 17, 402-406 (1979); translation from Algebra Logika 17, 611-618 (1978).

##### MSC:
 20F16 Solvable groups, supersolvable groups 20E36 Automorphisms of infinite groups 20E07 Subgroup theorems; subgroup growth 20F18 Nilpotent groups
Full Text:
##### References:
 [1] Yu. M. Gorchakov, ”Infinite Frobenius groups,” Algebra Logika,4, No. 1, 15–29 (1965). [2] Kourovka Notebook (Unsolved Problems of Group Theory) [in Russian], 5th edition, Novosibirsk (1976). [3] O. H. Kegel, ”Die Nilpotenz der Hp-Gruppen,” Math. Z.,75, No. 4, 373–376 (1961). · Zbl 0104.24904 · doi:10.1007/BF01211033 [4] J. G. Thompson, ”Finite groups with fixed-point-free automorphisms of prime order,” Proc. Nat. Acad. Sci. USA,45, 578–581 (1959). · Zbl 0086.25101 · doi:10.1073/pnas.45.4.578 [5] D. R. Hughes and J. G. Thompson, ”The H-problem and the structure of H-groups,” Pac. J. Math.,9, 1097–1101 (1959). · Zbl 0098.25201 [6] G. Higman, ”Groups and rings having automorphisms without nontrivial fixed elements,” J. London Math. Soc.,32, 321–334 (1957). · Zbl 0079.03203 · doi:10.1112/jlms/s1-32.3.321 [7] V. A. Kreknin and A. I. Kostrikin, ”Lie algebras with a regular automorphism,” Dokl. Akad. Nauk SSSR,149, No. 2, 249–251 (1963). · Zbl 0125.28902 [8] V. A. Kreknin, ”On the solvability of Lie algebras with a regular automorphism of finite order,” Dokl. Akad. Mauk SSSR,150, No. 3, 467–469 (1963). · Zbl 0134.03604 [9] P. Hall, ”On the finiteness of certain soluble groups,” Proc. London Math. Soc.,9, 595–622 (1959). · Zbl 0091.02501 · doi:10.1112/plms/s3-9.4.595 [10] P. J. Higgins, ”Lie rings satisfying the Engel condition,” Proc. Cambridge Phil. Soc.,50, No. 1, 8–15 (1954). · Zbl 0055.02601 · doi:10.1017/S0305004100029017 [11] G. Baumslag, ”Some aspects of groups with unique roots,” Acta Math.,104, Nos. 3–4, 217–303 (1960). · Zbl 0178.34901 · doi:10.1007/BF02546390 [12] C. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience, New York (1962). · Zbl 0131.25601
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