The Automorphism Tower Problem

The Automorphism Tower

It is well-known that the automorphism towers of infinite centre-less groups of cardinality κ terminate in less than (2 κ) + steps. But an easy counting argument shows that (2 κ) + is not the best possible bound. However, in this paper, we will show that it is impossible to find an explicit better bound using ZF C.

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 Groups Acting on Rooted Trees

The group of isometries Aut(Tn) of a rooted n-ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in Aut(Tn). This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group G studied by R. Grigorchuk, and the group Γ̈ studied by N. Gupta and the second author. In this paper, we pursue t...

The automorphism tower of a centerless group without Choice