Automorphisms of finite groups

To the memory of Anatolii Illarionovich Shirshov.     

Automorphisms and fusion in finite groups

We study how the fixed point subgroup of an automorphism influences the structure of a group.     

Automorphisms of finite groups of bounded rank

LetG be a finite group admitting an automorphismα withm fixed points. Suppose every subgroup ofG isr-generated. It is shown that (1)G has a characteristic soluble subgroupH whose index is bounded in terms ofm andr, and (2) if the orders ofα andG are coprime, then the derived length ofH is also bounded in terms ofm andr. To Professor John Thompson, in honor of his outstanding achievements     

Automorphisms of direct products of finite groups

This paper shows that if H and K are finite groups with no common direct factor and G = H × K, then the structure and order of Aut G can be simply expressed in terms of Aut H, Aut K and the central homomorphism groups Hom (H, Z(K)) and Hom (K, Z(H)). Received: 18 April 2005; revised: 9 June 2005     

Automorphisms of -subshifts of finite type

Let (Σ, σ) be a ^d-subshift of finite type. Under a strong irreducibility condition (strong specification), we show that Aut(Σ) contains any finite group. For ^d-subshift of finite type without strong specification, examples show that topological mixing is not sufficient to give any finite group in the automorphism group in general: in particular, End(Σ) may be an abelian semigroup. For an example of a topologically mixing ²-subshift of finite type, the endomorphism semigroup and automorphism group are computed explicitly. This subshift has periodic-point permutations that do not extend to automorphisms.     

Automorphisms fixing elements of prime order in finite groups

Let σ be an automorphism of a finite group G and suppose that σ fixes every element of G that has prime order or order 4. The main result of this paper shows that the structure of the subgroup H=[G, σ] is severely limited in terms of the order n of σ. In particular, H has exponent dividing n and it is nilpotent of class bounded in terms of n.     

Automorphisms of groups

The survey presents classical assertions due to Nielsen, Whitehead, and others, well-known theorems on automorphisms included in monographs on group theory, and recent results in this area. Attention is focused on the progress in automorphism groups theory for free, solvable, modular, and profinite groups. New tools of investigation using graphs and geometrical ideas are also discussed.     

Automorphisms of metacyclic groups

A metacyclic group H can be presented as 〈α,β: αⁿ = 1, β^m = α^t, βαβ^?1 = α^r〉 for some n, m, t, r. Each endomorphism σ of H is determined by \(\sigma(\alpha)=\alpha^{x_1}\beta^{y_1}, \sigma(\beta)=\alpha^{x_2}\beta^{y_2}\) for some integers x₁, x₂, y₁, y₂. We give sufficient and necessary conditions on x₁, x₂, y₁, y₂ for σ to be an automorphism.     

Linearity of artin groups of finite type

Recent results on the linearity of braid groups are extended in two ways. We generalize the Lawrence Krammer representation as well as Krammer’s faithfulness proof for this linear representation to Artin groups of finite type. This paper was written during a stay of the first author at Caltech. He wants to thank the institute for its hospitality. Shortly after we circulated a preprint of the present paper, we learned that Digne had obtained similar results.     

On the finite groups of supersoluble type

We study the properties of finite groups in which every Sylow subgroup can be connected to the group by a chain of subgroups of prime indices. We establish the solubility of this type of groups. We prove that the class of all finite groups with this property of Sylow subgroups is a saturated hereditary formation. For these groups we find some analogs of the available theorems on the products of normal supersoluble subgroups.     

Artin groups of finite type are biautomatic

Partially supported by NSF grant DMS-8905378