Abstract
Let a group Gsatisfy condition A: for every positive integer n the number of all subgroups of the group G of index n is finite. We prove that if G is virtually residually finite p-group for some prime p, then the automorphism group of the group G is virtually residually finite p-group. A similar result is obtained for a split extension of the group G by virtually residually finite p-group. Moreover, we prove that if the group G is a virtually residually finite nilpotent π-group for some finite set π of primes, then the automorphism group of the group G and the split extension of the group G by a virtually residually finite nilpotent π-group are virtually residually finite nilpotent π-groups.
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Original Russian Text © D.N. Azarov, 2015, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2015, No. 8, pp. 3–13.
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Azarov, D.N. Residual properties of automorphisms groups and split extensions. Russ Math. 59, 1–8 (2015). https://doi.org/10.3103/S1066369X15080010
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DOI: https://doi.org/10.3103/S1066369X15080010