Salut, André, I'm ashamed how long it took me to come to this realization, but you were absolutely correct in your surmise that, when one lets ...
... E' be the set obtained from E by adding copy p' of p. There are two embeddings u,u':E-->E', the first u is the inclusion of E in E', and the second u' is defined by putting u'(p)=p' and u'(x)=x for x different than p.
... the resulting u' is the same as the result of conjugating all the values of u by the transposition t that exchanges p with p'. Thus, the ...
... pair of homomorphisms h,h':E!-->E'! the equaliser of which is the stabiliser S(p) of p in E!.
... that arises is exactly the same as the pair Sammy's argument would adduce, and the only respect in which ...
This last argument seems to differ from the argument you have presented.
is that for Sammy it was enough to observe that h and h' differ SOMEWHERE when E is more than just {p}, while what you observe is rather more, namely, that h and h' actually differ EVERYWHERE other than on S(p), i.e., that the ONLY place where h and h' do NOT differ is S(p) :-) .
Am I making an error?
Only in thinking that seeming difference makes any real difference :-) . I hope this clarifies matters, and I think it's worth showing categories@mta. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]