Every group has a terminating transfinite automorphism tower

The Automorphism Tower of a Free Group

We prove that the automorphism group of any non-abelian free group F is complete. The key technical step in the proof: the set of all conjugations by powers of primitive elements is first-order parameter-free definable in the group Aut(F ). Introduction In 1975 J. Dyer and E. Formanek [2] had proved that the automorphism group of a finitely generated non-abelian free group F is complete (that i...

The Height of the Automorphism Tower of a Group Sh810

For a group G with trivial center there is a natural embedding of G into its automorphism group, so we can look at the latter as an extension of the group. So an increasing continuous sequence of groups, the automorphism tower, is defined, the height is the ordinal where this becomes fixed, arriving to a complete group. We show that for many such κ there is such a group of cardinality κ which i...

The automorphism tower of a centerless group without Choice

The Automorphism Tower of a Centerless Group (mostly) without Choice

For a centerless group G, we can define its automorphism tower. We define G: G = G, G = Aut (G) and for limit ordinals G = ⋃ α<δ G. Let τG be the ordinal when the sequence stabilizes. Thomas’ celebrated theorem says τG < ( 2 + and more. If we consider Thomas’ proof too set theoretical, we have here a shorter proof with little set theory. However, set theoretically we get a parallel theorem with...

Every Shift Automorphism Variety Has an Infinite Subdirectly Irreducible Member

A shift automorphism algebra is one satisfying the conditions of the shift automorphism theorem, and a shift automorphism variety is a variety generated by a shift automorphism algebra. In this paper, we show that every shift automorphism variety contains a countably infinite subdirectly irreducible algebra. 2000 Mathematics subject classification: primary 03C05; secondary 08B05, 08B26.