Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and x, y ∈ X (x 6= y) joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by ∆(G). The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut(∆(G)) is abelian if and only if ...

Let F be the finite field of order q, q = p with p prime. It is commonly atribute to J.P. Serre the fact that any curve F-covered by the Hermitian curve Hq+1 : y = x + x is also F-maximal. Nevertheless, the converse is not true as the GiuliettiKorchmáros example shows provided that q > 8 and h ≡ 0 (mod 3). In this paper, we show that if an F-maximal curve X of genus g ≥ 2 where q = p is such th...

For odd primes p, we examine Ĥ(Aut(F2(p−1));Z(p)), the Farrell cohomology of the group of automorphisms of a free group F2(p−1) on 2(p − 1) generators, with coefficients in the integers localized at the prime (p) ⊂ Z. This extends results in [9] by Glover and Mislin, whose calculations yield Ĥ(Aut(Fn);Z(p)) for n ∈ {p− 1, p} and is concurrent with work by Chen in [6] where he calculates Ĥ(Aut(F...

Let X be a compact connected Riemann surface of genus g ≥ 2, and let MDH be the rank one Deligne–Hitchin moduli space associated to X. It is known thatMDH is the twistor space for the hyper-Kähler structure on the moduli space of rank one holomorphic connections on X. We investigate the group Aut(MDH) of all holomorphic automorphisms of MDH. The connected component of Aut(MDH) containing the id...

Let Aut(O"A) denote the group of automorphisms of a subshift of finite type (XA'O"A) built from a primitive matrix A. We show that the sign-gyrationcompatibility-condition homomorphism SGCC A, m defined on Aut( 0" A) factors through the group Aut(s A) of automorphisms of the dimension group. This is used to find a mixing subshift of finite type with a permutation of fixed points that cannot be ...

We use the following rule of composition of automorphisms: if φ, ψ ∈ Aut(F2) and x ∈ F2 then φψ(x) = ψ(φ(x)). For x ∈ F2 denote by St(x) the stabilizer of x in Aut(F2). For a subset X of a group denote by 〈X〉 the subgroup generated by X. Denote [x, y] = x−1y−1xy, x = y−1xy. For g ∈ F2 denote by ĝ the automorphism induced by the conjugation by g: ĝ(x) = g−1xg, x ∈ F2. Let − : Aut(F2) → GL2(Z) be...

We give upper bounds on the order of the automorphism group of a simple graph In this note we present some upper bounds on the order of the automorphism group of a graph, which is assumed to be simple, having no loops or multiple edges. Somewhat surprisingly, we did not find such bounds in the literature and the goal of this paper is to fill this gap. As a matter of fact, implicitly such bounds...

The soil microbiota plays an extremely important role in the growth and survival of plants. presence some microorganisms can positively significantly impact tree species, which improve performance seedlings after planting for commercial purposes and/or ecosystem restoration. present study aimed to evaluate initial Hancornia speciosa Brosimum gaudichaudii associated with from parent inoculated a...

V = {f | f is a homogeneous quintic polynomial of Z0, · · · , Z4}. We can verify that dimV = 126. Thus for any t ∈ P (V ) = CP , t is represented by a hypersurface. However, if two hypersurfaces differ by an element in Aut(CP ), then they are considered to be the same. Let D be the divisor in CP 125 characterizing the singular hypersurfaces in CP . Then the moduli space of X is M = CP \D/Aut(CP ).

We describe under various conditions abelian subgroups of the automorphism group Aut(Tn) of the regular n-ary tree Tn, which are normalized by the n-ary adding machine τ = (e, . . . , e, τ)στ where στ is the n-cycle (0, 1, . . . , n− 1). As an application, for n = p a prime number, and for n = 4, we prove that every soluble subgroup of Aut(Tn), containing τ is an extension of a torsion-free met...