In this paper, we consider the group Aut(Q, ≤) of order-automorphisms of the rational numbers, proving a result analogous to a theorem of Galvin's for the symmetric group. In an announcement, Khélif states that every countable subset of Aut(Q, ≤) is contained in an N-generated subgroup of Aut(Q, ≤) for some fixed N ∈ N. We show that the least such N is 2. Moreover, for every countable subset of...

Let M be a closed symplectic manifold, with symplectic form ω. A symplectic automorphism is a diffeomorphism φ : M → M such that φω = ω. We equip the group Aut(M) = Aut(M,ω) of all such maps with the C-topology. Like the whole diffeomorphism group, this is an infinite-dimensional Lie group in a very loose sense: it has a well-defined Lie algebra, which consists of closed oneforms on M , but the...

Let Fn be the free group of rank n and let Aut +(Fn) be its special automorphism group. For an epimorphism π : Fn → G of the free group Fn onto a finite group G we call Γ (G, π) = {φ ∈ Aut+(Fn) | πφ = π} the standard congruence subgroup of Aut+(Fn) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ(G, π) for finite abelian groups G. Moreover, we show that if G is...

Let X be a homogeneous compact complex manifold, and let Aut(X) be the complex Lie group of holomorphic automorphisms of X. Examples show that dim Aut(X) can grow exponentially in n = dimX. In this note it is shown that dim Aut(X) ≤ n − 1 + (2n− 1 n− 1 ) when n ≥ 3. Thus, dim Aut(X) is at most exponential in n. The proof relies on an upper bound for the dimension of the space of sections of the...

In this article we introduce and study the concept of characteristic degree of a subgroup in a finite group. We define the characteristic degree of a subgroup H in a finite group G as the ratio of the number of all pairs (h,α) ∈ H×Aut(G) such that h^α∈H, by the order of H × Aut(G), where Aut(G) is the automorphisms group of G. This quantity measures the probability that H can be characteristic ...

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...

To every graph (or digraph) A, there is an associated automorphism group Aut(A). Frucht’s theorem asserts the converse association; that for any finite group G there is a graph (or digraph) A for which Aut(A) ∼= G. A new operation on digraphs was introduced recently as an aid in solving certain questions regarding cancellation over the direct product of digraphs. Given a digraph A, its factoria...

A labeling of the vertices of a graph G, φ : V (G) → {1, . . . , r}, is said to be r-distinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by D(G), is the minimum r such that G has an r-distinguishing labeling. The distinguishing number of the complete graph on t vertices is t. In contrast, we prove (i) given any...

For a subshift over a finite alphabet, a measure of the complexity of the system is obtained by counting the number of nonempty cylinder sets of length n. When this complexity grows exponentially, the automorphism group has been shown to be large for various classes of subshifts. In contrast, we show that subquadratic growth of the complexity implies that for a topologically transitive shift X,...