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.
February 27, 2006 version, 8 pages, available as
Return to Warren Dicks' publications.