Geodesic
Polycyclic groups with automorphisms of order four
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 575-582
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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.
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.
DOI : 10.1007/s10587-016-0276-8
Classification : 20E36
Keywords: polycyclic group; regular automorphism; surjectivity 
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Xu, Tao; Zhou, Fang; Liu, Heguo. Polycyclic groups with automorphisms of order four. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 575-582. doi: 10.1007/s10587-016-0276-8

[1] Burnside, W.: Theory of Groups of Finite Order. Dover Publications New York (1955). | MR | Zbl

[2] Endimioni, G.: On almost regular automorphisms. Arch. Math. 94 (2010), 19-27. | DOI | MR | Zbl

[3] Endimioni, G., Moravec, P.: On the centralizer and the commutator subgroup of an automorphism. Monatsh. Math. 167 (2012), 165-174. | DOI | MR | Zbl

[4] Gorenstein, D.: Finite Groups. Harper's Series in Modern Mathematics, Harper and Row, Publishers New York (1968). | MR | Zbl

[5] Higman, G.: Groups and rings having automorphisms without non-trivial fixed elements. J. Lond. Math. Soc. 32 (1957), 321-334. | DOI | MR

[6] Kovács, L. G.: Group with regular automorphisms of order four. Math. Z. 75 (1961), 277-294. | DOI | MR

[7] Lennox, J. C., Robinson, D. J. S.: The Theory of Infinite Soluble Groups. Oxford Science Publications Oxford (2004). | MR | Zbl

[8] Neumann, B. H.: Groups with automorphisms that leave only the neutral element fixed. Arch. Math. 7 (1956), 1-5. | DOI | MR | Zbl

[9] Robinson, D. J. S.: A Course in the Theory of Groups. Springer New York (1996). | MR

[10] Shmel'kin, A. L.: Polycyclic groups. Sib. Math. J. Russian 9 (1968), 234-235. | Zbl

[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 | MR | Zbl

[13] Thompson, J. G.: Finite group with fixed-point-free automorphisms of prime order. Proc. Natl. Acad. Sci. 45 (1959), 578-581. | DOI | MR

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