Normal Closure of Finite Subgroups of Aut(Fn) and Out(Fn)
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posted on 2025-04-23, 14:45authored byJiayi Shen
For the integer n at least 3, let G be a nontrivial finite subgroup of Aut(Fn) with the order of G not a power of 2. We prove that the normal closure N(G) is Aut(Fn) if G is contained in SAut(Fn) and N(G) is Aut(Fn) otherwise. When the order of G is a power of 2, we have a partial theorem. Similarly, let G' be a nontrivial finite subgroup of Out(Fn) with the order of G' not a power of 2. Then the normal closure N(G') is SOut(Fn) if G' is contained in SOut(Fn) and N(G') is Out(Fn) otherwise. When the order of G' is a power of 2, we have a partial theorem as well.