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Keywords:
polycyclic group; regular automorphism; surjectivity
polycyclic group; regular automorphism; surjectivity
Summary:
In this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann's result, and prove that if $\alpha $ is an automorphism of order four of a polycyclic group $G$ and the map $\varphi \colon G\rightarrow G$ defined by $g^{\varphi }=[g,\alpha ]$ is surjective, then $G$ contains a characteristic subgroup $H$ of finite index such that the second derived subgroup $H''$ is included in the centre of $H$ and $C_{H}(\alpha ^{2})$ is abelian, both $C_{G}(\alpha ^{2})$ and $G/[G,\alpha ^{2}]$ are abelian-by-finite. These results extend recent and classical results in the literature.
References:
[1] Burnside, W.: Theory of Groups of Finite Order. Dover Publications New York (1955). MR 0069818 | Zbl 0064.25105
[2] Endimioni, G.: On almost regular automorphisms. Arch. Math. 94 (2010), 19-27. DOI 10.1007/s00013-009-0084-6 | MR 2581329 | Zbl 1205.20041
[3] Endimioni, G., Moravec, P.: On the centralizer and the commutator subgroup of an automorphism. Monatsh. Math. 167 (2012), 165-174. DOI 10.1007/s00605-011-0298-0 | MR 2954523 | Zbl 1270.20031
[4] Gorenstein, D.: Finite Groups. Harper's Series in Modern Mathematics, Harper and Row, Publishers New York (1968). MR 0231903 | Zbl 0185.05701
[5] Higman, G.: Groups and rings having automorphisms without non-trivial fixed elements. J. Lond. Math. Soc. 32 (1957), 321-334. DOI 10.1112/jlms/s1-32.3.321 | MR 0089204
[6] Kovács, L. G.: Group with regular automorphisms of order four. Math. Z. 75 (1961), 277-294. DOI 10.1007/BF01211026 | MR 0123613
[7] Lennox, J. C., Robinson, D. J. S.: The Theory of Infinite Soluble Groups. Oxford Science Publications Oxford (2004). MR 2093872 | Zbl 1059.20001
[8] Neumann, B. H.: Groups with automorphisms that leave only the neutral element fixed. Arch. Math. 7 (1956), 1-5. DOI 10.1007/BF01900516 | MR 0074413 | Zbl 0070.02203
[9] Robinson, D. J. S.: A Course in the Theory of Groups. Springer New York (1996). MR 1357169
[10] Shmel'kin, A. L.: Polycyclic groups. Sib. Math. J. Russian 9 (1968), 234-235. Zbl 0203.32602
[11] Tao, X., Heguo, L.: Polycyclic groups admitting an automorphism of prime order. Submitted to Ukrainnian Math. J.
[12] Tao, X., Heguo, L.: On regular automorphisms of soluble groups of finite rank. Chin. Ann. Math. A35 (2014), Chinese 543-550, doi 10.3103/S0898511114040036. DOI 10.3103/S0898511114040036 | MR 3290005 | Zbl 1324.20023
[13] Thompson, J. G.: Finite group with fixed-point-free automorphisms of prime order. Proc. Natl. Acad. Sci. 45 (1959), 578-581. DOI 10.1073/pnas.45.4.578 | MR 0104731
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