Endomorphisms of polycyclic-by-finite groups. (English) Zbl 1197.20025
In the article under review some conditions for an endomorphism of a polycyclic (by-finite) group to be an automorphism are investigated. The main results obtained are the following:
Theorem 1. Let \(G\) be a polycyclic group, and let \(\varphi\) be an endomorphism of \(G\). If the restriction of \(\varphi\) to the centre \(Z(\text{Fit\,}G)\) of the Fitting subgroup of \(G\) (i.e., the subgroup generated by all the normal nilpotent subgroups of \(G\)) is an automorphism, then also \(\varphi\) is an automorphism.
Theorem 2. Let \(G\) be a polycyclic-by-finite group, and let \(\varphi\) be a monic endomorphism of \(G\). If the restriction of \(\varphi\) to the centre \(Z(\text{Fit\,}G)\) of the Fitting subgroup of \(G\) is an automorphism, then also \(\varphi\) is an automorphism.
The above theorems extend (independent) results of D. R. Farkas and the author in the 1980s.
Theorem 1. Let \(G\) be a polycyclic group, and let \(\varphi\) be an endomorphism of \(G\). If the restriction of \(\varphi\) to the centre \(Z(\text{Fit\,}G)\) of the Fitting subgroup of \(G\) (i.e., the subgroup generated by all the normal nilpotent subgroups of \(G\)) is an automorphism, then also \(\varphi\) is an automorphism.
Theorem 2. Let \(G\) be a polycyclic-by-finite group, and let \(\varphi\) be a monic endomorphism of \(G\). If the restriction of \(\varphi\) to the centre \(Z(\text{Fit\,}G)\) of the Fitting subgroup of \(G\) is an automorphism, then also \(\varphi\) is an automorphism.
The above theorems extend (independent) results of D. R. Farkas and the author in the 1980s.
Reviewer: Alessio Russo (Napoli)
MSC:
| 20E36 | Automorphisms of infinite groups |
| 20F19 | Generalizations of solvable and nilpotent groups |
| 20F16 | Solvable groups, supersolvable groups |
Keywords:
endomorphisms; polycyclic groups; automorphisms; centre of Fitting subgroup; polycyclic-by-finite groupsReferences:
| [1] | Farkas D.R. (1982). Endomorphisms of polycyclic groups. Math. Zeit. 181: 567–574 · Zbl 0495.20014 · doi:10.1007/BF01182394 |
| [2] | Hall, P.: Lectures on Nilpotent Groups, Canad. Math. Congress, Univ. Alberta 1957; reissued as Queen Mary College Math. Notes, London 1969. Also in The Collected Works of Philip Hall, Oxford University Press, New York (1988) |
| [3] | Robinson D.J.S. (1972). Finiteness Conditions and Generalized Soluble Groups. Springer, Berlin · Zbl 0243.20032 |
| [4] | Wehrfritz B.A.F. (1983). Endomorphisms of polycyclic groups. Math. Zeit. 184: 97–99 · doi:10.1007/BF01162008 |
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