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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

MSC:

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