✍ Scribed by A. Martino; E. Ventura
No coin nor oath required. For personal study only.
Let n be an arbitrary cardinal, and let F n be a free group of rank n. The fixed subgroup of an endomorphism ψ of F n is the subgroup of elements in F n fixed by ψ. In this paper, the relationship between the family of fixed subgroups of endomorphisms of F n and the family of fixed subgroups of automorphisms of F n will be studied. We prove that these two families of subgroups do not coincide for n 3, by showing an infinite sequence of explicit examples of retracts of F n -and so, fixed subgroups of endomorphisms of F n -which are not fixed subgroups of any automorphism of F n .
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