Automorphism sequences of integer unimodular groups
Joan Landman Dyer · Illinois Journal of Mathematics · 1978
Let /(G) denote the automorphism group of the group G, and let I" G -/(G) be the homomorphism which assigns to g e G the inner automorphism l(g): x -gxg -(all x s G).is procedure may be iterated to v rise to an automorphism sequence G (G) ((G))= 2(G) '".Such a sequence stabilizes in finitely many steps if the maps I" '(G) '* (G) are isomorphisms for all sufficiently large integers i; that is, (G) has a trivial center and only inner automorphisms for all sufficiently large i.Such roups are reded complete.Finite stability need not occur, even when G is assumed to be linear.For th infinit dihedral roup D, ach " '(G) '+ '(G) is a monomorphism with + '(G)/I((G))of order two (Hulse [7]).The main rsult of this paper is" THEOREM A. The automorphism sequences of the groups SL(n, Z)and GL(n, Z) stabilize in finitely many steps.When G is SL(n, Z) or GL(n, Z), the automorphism group /(G) is known (Hua and Reiner [5], Wan [17]).Moreover, G has almost all automorphisms inner, and almost has a trivial center.Thus the conclusion of Theorem A is a natural one, and the automorphism sequences of these groups might be ex- pected to stabilize very quickly.However the situation is surprisingly com- plicated for the general linear group when n is even, as well as for the special linear group when n 2.We first establish: THEOREM B. For n >_ 2, .q/(PGL(n,Z)) is complete" that is I" ./(PGL(n,Z)-, ./2(PGL(n,Z))is an isomorphism.