Every group has a terminating transfinite automorphism tower

Iteratively taking the automorphism group of any group leads, transfinitely, to a fixed point.

     

The automorphism tower problem II

We prove that the automorphism tower of every infinite centreless groupG of cardinality κ terminates in less than (2^κ)⁺ steps. We also show that it is consistent withZFC that the automorphism tower of every infinite centreless groupG of regular cardinality κ actually terminates in less than 2^κ steps. Research partially supported by NSF Grants.     

The automorphism group of a coded system

We give a general construction of coded systems with an automorphism group isomorphic to where is any preassigned group which has a ``continuous block presentation' (the isomorphism will map the shift to . Several applications are given. In particular, we obtain automorphism groups of coded systems which are abelian, which are finitely generated and one which contains . We show that any group which occurs as a subgroup of the automorphism group of some subshift with periodic points dense already occurs for some synchronized system.

     

The automorphism group of a circle geometry

The following is proved: Theorem. Let n be an integer, n ? 2. Let F be a field with at least Max(n, 3) distinct elements and let K be a separable extension field of dimension n over F. Then the group of all automorphisms of the n-dimensional circle geometry CG(F, K) is the group induced on the 1-dimensional vector subspaces over K of the underlying 2-dimensional vector space V over K by the semi-linear permutations of V over K for which the associated automorphisms of K induce automorphisms of F.With the hope of broadening the interest in the foundations of geometry, the author attempts to give anough detail so that, without consulting the literature, any attentive algebraist can understand clearly both the content of the theorem and the pattern of its proof and can identify easily any gaps in his knowledge which might prevent him from viewing this paper as a contribution to geometry. Adequate references are given.     

The automorphism group of a product of hypergraphs

A theorem of G. Sabidussi (1959, Duke Math. J. 26, 693–696) gives necessary and sufficient conditions for the automorphism group of the wreath product of two graphs to be the wreath product of their respective automorphism groups. In this paper we define a wreath product of hypergraphs and prove a theorem extending that of Sabidussi.     

The automorphism group of a class of semi-groups

Let (S,·) be a semi-group having the following properties: (1)S=∪S _α where α is in some index setI andS _α are subgroups isomorphic to each other, (2)S _α∩S _β=Ø, a void set for α≠β and (3) the identity ofS _α is a left identity ofS for each α inI. Then the automorphism group Aut (S) ofS is studied from the point of category theory. It is proved that Aut (S) is determined by Aut (S _α) and right multiplications by the identities of groupsS _α.     

The automorphism group of a nonsplit metacyclic p-group

This note considers a finite group G = HK, which is a product of a subgroup H and a normal subgroup K, and determines subgroups of Aut G. The special case when G is a nonsplit metacyclic p-group, where p is odd, is then considered and the structure of its automorphism group Aut G is given. Received: 13 September 2007, Revised: 22 November 2007     

The automorphism group of a generalized extraspecial p-group

In this paper, the automorphism group of a generalized extraspecial p-group G is determined, where p is a prime number. Assume that |G| = p 2n+m and |ζG| = p m , where n 1 and m 2. (1) When p is odd, let Aut G G = {α∈ AutG | α acts trivially on G }. Then Aut G G⊿AutG and AutG/Aut G G≌Z p-1 . Furthermore, (i) If G is of exponent p m , then Aut G G/InnG≌Sp(2n, p) × Z p m-1 . (ii) If G is of exponent p m+1 , then Aut G G/InnG≌ (K Sp(2n-2, p))×Z p m-1 , where K is an extraspecial p-group of order p 2n-1 . In particular, Aut G G/InnG≌ Z p × Z p m-1 when n = 1. (2) When p = 2, then, (i) If G is of exponent 2 m , then AutG≌ Sp(2n, 2) × Z 2 × Z 2 m-2 . In particular, when n = 1, |AutG| = 3 · 2 m+2 . None of the Sylow subgroups of AutG is normal, and each of the Sylow 2-subgroups of AutG is isomorphic to H K, where H = Z 2 × Z 2 × Z 2 × Z 2 m-2 , K = Z 2 . (ii) If G is of exponent 2 m+1 , then AutG≌ (I Sp(2n-2, 2)) × Z 2 × Z 2 m-2 , where I is an elementary abelian 2-group of order 2 2n-1 . In particular, when n = 1, |AutG| = 2 m+2 and AutG≌ H K, where H = Z 2 × Z 2 × Z 2 m-1 , K = Z 2 .     

The automorphism group of a rigid affine variety

An irreducible algebraic variety X is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group of a rigid affine variety contains a unique maximal torus . If the grading on the algebra of regular functions defined by the action of  is pointed, the group  is a finite extension of . As an application, we describe the automorphism group of a rigid trinomial affine hypersurface and find all isomorphisms between such hypersurfaces.     

The automorphism group of a split metacyclic p-group

This paper finds the order, structure and presentation for the automorphism group of a split metacyclic p-group, where p is odd. Received: 8 February 2006     

The automorphism group of the augmented equiaffine group

The automorphism group of the group generated by all the affine reflections in a Desarguesian plane is isomorphic to the full collineation group of the plane.The author wishes to express his thanks to J. Wilker of the University of Toronto for suggesting a simplification and to him as well as C. Fisher of the University of Regina for their encouragement.     

The automorphism tower of groups acting on rooted trees

The group of isometries of a rooted -ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in . This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group studied by R. Grigorchuk, and the group studied by N. Gupta and the second author.

In this paper, we pursue the larger theme of towers of automorphisms of groups of tree isometries such as and . We describe this tower for all subgroups of which decompose as infinitely iterated wreath products. Furthermore, we fully describe the towers of and .

More precisely, the tower of is infinite countable, and the terms of the tower are -groups. Quotients of successive terms are infinite elementary abelian -groups.

In contrast, the tower of has length , and its terms are -groups. We show that is an elementary abelian -group of countably infinite rank, while .