Endomorphisms of two-generated metabelian groups that induce the identity modulo the derived subgroup. (English) Zbl 0693.20038

We prove that metabelian, two-generated groups admit all possible endomorphisms that induce the identity modulo the derived subgroup. We characterize nilpotent groups in this class as those for which all these endomorphisms are actually automorphisms. We show that for these groups the lower central series of the group of IA-automorphisms, that is, of those automorphisms that induce the identity on the quotient modulo the derived subgroup, coincides with the lower central series of the group of inner automorphisms. We use the ring-theoretic techniques that have been developed by various authors for metabelian groups.
Reviewer: A.Caranti


20F16 Solvable groups, supersolvable groups
20E36 Automorphisms of infinite groups
20F14 Derived series, central series, and generalizations for groups
20F18 Nilpotent groups
Full Text: DOI


[1] S. Bachmuth, G. Baumslag, J. Dyer andH. Y. Mochizuki, Automorphism groups of two generator metabelian groups. J. London Math. Soc.36, 393-406 (1987). · Zbl 0638.20022
[2] N.Bourbaki, ?l?ments de Math?matique XXX, Alg?bre Commutative, Chapitres 5-6. Paris 1964.
[3] A.Caranti and C. M.Scoppola, A remark on the orders ofp-groups that are automorphism groups. To appear in Boll. Un. Mat. Ital. · Zbl 0703.20017
[4] C. K. Gupta, IA-automorphisms of two generator metabelian groups. Arch. Math.37 106-112 (1981). · Zbl 0472.20014
[5] B.Huppert, Endliche Gruppen I. Berlin-Heidelberg-New York 1967. · Zbl 0217.07201
[6] N.Jacobson, Basic Algebra I. San Francisco 1974; Basic Algebra II. San Francisco 1980. · Zbl 0284.16001
[7] Yu. V. Kuz’min, Inner endomorphisms of abelian groups, (Russian). Sibirsk. Mat. Zh.16, 736-774 (1975); English translation: Siberian Math. J.16, 563-568 (1975).
[8] G.Szekeres, Metabelian groups with two generators. In: Proc. Internat. Conf. Theory of Groups, Austral. Nat. Univ. Canberra, August 1965, 323-346, New York 1967.
[9] B. A. F.Wehrfritz, Infinitite Linear Groups, Berlin-Heidelberg-New York 1973. · Zbl 0266.20052
[10] H.Zassenhaus, The Theory of Groups, Second Edition. New York 1958. · Zbl 0087.01401
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.