Two examples in the Galois theory of free groups

Let F be a free group, and let H be a subgroup of F. The 'Galois monoid' EndH (F) consists of all endomorphisms of F which fix every element of H; the 'Galois group' AutH (F) consists of all automorphisms of F which fix every element of H. The End (F)-closure and the Aut (F)-closure of H are the fixed subgroups, Fix (EndH (F)) and Fix (AutH (F)), respectively. Martino and Ventura considered examples whereFix (AutH (F)) ≠ Fix (EndH (F)) = H . We obtain, for two of their examples, explicit descriptions of EndH (F), AutH (F), and Fix (AutH (F)), and, hence, give much simpler verifications that Fix (AutH (F)) ≠ Fix (EndH (F)), in these cases. © 2006 Elsevier Inc. All rights reserved.