Why do free monoids have a "trivial" automorphism group and free groups don't?

Solution 1:

Automorphisms of free algebraic structures are well-studied in examples, but many problems are open, and of course there is no general answer to the question when all automorphisms are just permutations of the free generators.

For groups, see "Combinatorial group theory" (Lyndon, Schupp). For lie algebras and various other examples, see "Free rings and their relations" (Cohn). The title "Automorphisms of a free associative algebra of rank $2$" (Czerniakiewicz) is self-explanatory, similarly "The automorphisms of the free algebra with two generators" (Makar-Limanov) and "On the automorphism group of $k[x,y]$" (Nagata). The group of automorphisms of $k[x,y,z]$ is in current research; see for example "Polynomial automorphisms and invariants" (van den Essen, Peretz) and "The tame and the wild automorphisms of polynomial rings in three variables" (Shestakov, Umirbaev).