A 2-arc Transitive Hexavalent Nonnormal Cayley Graph on A119

A Cayley graph Γ=Cay(G,S) is said to be normal if the base group G is normal in AutΓ. The concept of the normality of Cayley graphs was first proposed by M.Y. Xu in 1998 and it plays a vital role in determining the full automorphism groups of Cayley graphs. In this paper, we construct an example of a 2-arc transitive hexavalent nonnormal Cayley graph on the alternating group A119. Furthermore, we determine the full automorphism group of this graph and show that it is isomorphic to A120.

An 𝔽 p2 -maximal Wiman sextic and its automorphisms

In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X 6+Y 6+ℨ 6+(X 2+Y 2+ℨ 2)(X 4+Y 4+ℨ 4)−12X 2 Y 2 ℨ 2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S 5. We show that this holds also over every algebraically closed field 𝕂 of characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman’s 𝔽192 -maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192 .

p-Groups of automorphisms of compact non-orientable Riemann surfaces

We study p-groups of automorphisms of compact non-orientable Riemann surfaces of topological genus $$g\ge 3$$ g ≥ 3 . We obtain upper bounds of the order of such groups in terms of p, g and the minimal number of generators of the group. We also determine those values of g for which these bounds are sharp. Furthermore, the same kind of results are obtained when the p-group acts as the full automorphism group of the surface.

Hausdorff dimension of the second Grigorchuk group

We show that the Hausdorff dimension of the closure of the second Grigorchuk group is 43/128. Furthermore, we establish that the second Grigorchuk group is super strongly fractal and that its automorphism group equals its normalizer in the full automorphism group of the tree.

A class of continuous non-associative algebras arising from algebraic groups including

We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associative algebra A over that field. Two attractions of this construction are that (1) when G has type $E_8$ , the algebra A is obtained by adjoining a unit to the 3875-dimensional representation; and (2) it is effective, in that the product operation on A can be implemented on a computer. A description of the algebra in the $E_8$ case has been requested for some time, and interest has been increased by the recent proof that $E_8$ is the full automorphism group of that algebra. The algebras obtained by our construction have an unusual Peirce spectrum.

On plane curves given by separated polynomials and their automorphisms

Let 𝓒 be a plane curve defined over the algebraic closure K of a finite prime field 𝔽p by a separated polynomial, that is 𝓒 : A(Y) = B(X), where A(Y) is an additive polynomial of degree pn and the degree m of B(X) is coprime with p. Plane curves given by separated polynomials are widely studied; however, their automorphism groups are not completely determined. In this paper we compute the full automorphism group of 𝓒 when m ≢ 1 mod pn and B(X) = Xm. Moreover, some sufficient conditions for the automorphism group of 𝓒 to imply that B(X) = Xm are provided. Also, the full automorphism group of the norm-trace curve 𝓒 : X(qr – 1)/(q–1) = Yqr–1 + Yqr–2 + … + Y is computed. Finally, these results are used to show that certain one-point AG codes have many automorphisms.

ON A DESIGN FROM PRIMITIVE REPRESENTATIONS OF THE FINITE SIMPLE GROUPS

In this paper we present a design construction from primitive permutation representations of a finite simple group G. The group G acts primitively onthe points and transitively on the blocks of the design. The construction has this property that with some conditions we can obtain t-design for t >=2. We examine our design for fourteen sporadic simple groups. As a result we found a 2-(176,5,4) design with full automorphism group M22.

Automorphisms of prime-cube order group

A graph is called half-arc-transitive if its full automorphism group acts transitively on vertices and edges, but not on arcs. Let [Formula: see text] be a prime-cube order group. In this paper, we determined the structures of the automorphism groups of [Formula: see text], and the result is used to determine the classification of half-arc-transitive graphs of order [Formula: see text] with valency [Formula: see text].

Pentavalent arc-transitive Cayley graphs on Frobenius groups with soluble vertex stabilizer

A Cayley graph Γ is said to be arc-transitive if its full automorphism group AutΓ is transitive on the arc set of Γ. In this paper we give a characterization of pentavalent arc-transitive Cayley graphs on a class of Frobenius groups with soluble vertex stabilizer.

Automorphism Groups of Bouwer Graphs

Let k, m and n be three positive integers such that 2m ≡ 1 (mod n) and k ≥ 2. The Bouwer graph, which is denoted by B(k, m, n), is the graph with vertex set [Formula: see text] and two vertices being adjacent if they can be written as (a, b) and (a + 1, c), where either c = b or [Formula: see text] differs from [Formula: see text] in exactly one position, say the jth position, where [Formula: see text]. Every B(k, m, n) is a vertex- and edge-transitive graph, and Bouwer proved that B(k, 6, 9) is half-arc-transitive for every k ≥ 2. In 2016, Conder and Žitnik gave the classification of half-arc-transitive Bouwer graphs. In this paper, the full automorphism group of every B(k, m, n) is determined.