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Let G be a finitely generated group , then can we say that the group of automorphisms of G is also finitely generated .If yes what is the relation between the number of generators.If not under what condition the group of automorphism is finitely generated

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    $\begingroup$ It need not be finitely generated. Any countable group is the outer automorphism group of a fg group. $\endgroup$ Apr 8, 2014 at 14:00
  • $\begingroup$ @Benjamin: Do you have a reference for this interesting fact? $\endgroup$
    – abx
    Apr 8, 2014 at 14:08
  • $\begingroup$ unfortunately i have no reference. if instead of group of automorphism if we take group of central automorphisms then can we say that above result is true $\endgroup$
    – Ranjit
    Apr 8, 2014 at 14:11
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    $\begingroup$ Inna Bumagin and Daniel Wise proved it. sciencedirect.com/science/article/pii/S0022404904003111 $\endgroup$ Apr 8, 2014 at 14:37
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    $\begingroup$ The Baumslag-Solitar group $BS(2,4)$ has an infinitely generated automorphism group (D.J. Collins, F. Levin, Automorphisms and hopficity of certain Baumslag-Solitar groups, Arch. Math. 40 (1983), 385–400.) $\endgroup$
    – YCor
    Apr 8, 2014 at 17:14

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