From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: Free group on two generators a,b Date: 11 Apr 2001 20:24:27 GMT Newsgroups: sci.math Summary: Automorphism groups of free groups In article , Michael Berg wrote: >If G is the free group on two generators a,b what are the automorphosm >group of G and the commutator subgroup of G ? First, every subgroup of a free group is free; the commutator subgroup of a free group on n > 1 is a free group on countably many generators. As for the automorphism group, as you may know Aut(G) always includes the normal subgroup Inn(G) of inner automorphisms. Inn(G) is isomorphic to G/Z(G) and for a free group of rank greater than 1, Z(G) is trivial, so Inn(G) is also free. The interesting part of Aut(G) then comes from the "extra" automorphisms in Out(G) = Aut(G)/Inn(G). I cannot remember how large this group is for all free groups. Certainly Aut(G) contains the subgroup of signed permutations of the generators; this is the group (Z/2Z) wreath Sym(n) which is described in another thread of this newsgroup! None of these automorphisms is inner, as can be seen by looking at the induced actions on the abelianization G/[G,G]. So in your case, Aut(G) is at least as big as the semidirect product G.D where D is this group of signed permutations, in this case the dihedral group of order 8. Clearly there are many other _endomorphisms_ of G (just send a and b anywhere you like in G) but it takes a bit of care to construct other _automorphisms_ of G; one has to be able to describe the inverse map too. Just how much larger Aut(G) can be is limited by the action on G/[G,G]. Let me cite this review from MR: 23 #A1735 20.22 Chang, Bomshik The automorphism group of the free group with two generators. Michigan Math. J. 7 1960 79--81. A short, but not self-contained, proof of the following theorem is given: If $F$ is a free group of rank 2, then any automorphism of $F$, which induces the identity on $F/[F,F]$, is an inner automorphism. This theorem is due originally to J. Nielsen [Math. Ann. 78 (1918), 385--397]. Reviewed by K. Gruenberg (c) 1962, 2001, American Mathematical Society So Out(G) is precisely the image of Aut(G) in Aut(G/[G,G]). (Note: F=G !) Of course this quotient group G/[G,G] is the free _abelian_ group on two generators, i.e. Z x Z. Its automorphism group is GL_2(Z), the group of 2x2 matrices with determinant 1. It happens to be true that the subquotient PSL_2(Z) is freely generated by an element of order 2 and an element of order 3, and so in particular GL_2(Z) is then generated by the three elements [[1,0], [0,-1]], [[0,1],[-1,0]], and [[0,-1],[1,1]] (the latter two generating SL_2(Z) as a free product with amalgamation; but I digress!) It is easily seen that each of these three is in the image of Aut(G) -- respectively of the automorphisms sending a -> a b -> b^(-1) a -> b^(-1) b -> a a -> b b -> a^(-1) b The first two generate the dihedral group D of which I spoke earlier. The last is inverted by the automorphism sending a -> b a^(-1) b -> a and has a cube equal to the automorphism sending a -> a^(-1) c b -> b^(-1) c where c is the commutator a b^(-1) a^(-1) b . So the three automorphisms generate all of GL_2(Z) in Out(G). I'm not sure how much more detailed you need the answers to be. I know this sort of thing is well known to those who know it well :-) and is probably in books on combinatorial group theory (e.g. Magnus's). These three automorphisms together with the inner automorphisms by a and b generate all of Aut(G), and I seem to recall only three generators are really needed; I don't know what sorts of relations bind them. (Some, surely, since Aut(G) itself cannot be free, not being torsionfree.) Alternatively you can describe Aut(G) as a split extension G -> Aut(G) -> GL_2(Z) but I don't what you need to know about the extension. I don't think it splits. Similar ideas apply to automorphisms of free groups on other finite numbers of generators (more than 1 !) but I don't recall the details. dave