Laura Ciobanu and Warren Dicks,
Two examples in the Galois theory of free groups.
J. Algebra, 305 (2006), 540-547.

Abstract:  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 where
                              Fix (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 simpler verifications that
Fix (AutH (F )) ≠ Fix (EndH (F )), in these cases.

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