Solvable Automorphism Group Research Articles

Four infinite families of chiral 3-polytopes of type {4,8} with solvable automorphism groups

We construct four infinite families of chiral 3-polytopes of type {4,8}, with 1024m4, 2048m4, 4096m4 and 8192m4 automorphisms for every positive integer m, respectively. The automorphism groups of these polytopes are solvable groups, and when m is a power of 2, they provide examples with automorphism groups of order 2n where n≥10. (On the other hand, no chiral polytopes of type {4,8} exist for n≤9.) In particular, our families give a partial answer to a problem proposed by Schulte and Weiss (2006) [15] and a problem proposed by Pellicer (2012) [13].

More on chiral polytopes of type \{4, 4, …, 4\} with~solvable automorphism groups

Abstract In a 2021 paper, Conder et al. constructed two infinite families of chiral 4-polytopes of type { 4 , 4 , 4 } \{4,4,4\} with solvable automorphism groups. Here we present a general construction for chiral polytopes of type { 4 , 4 , … , 4 } \{4,4,\dots,4\} with rank 4, 5 and 6, which are obtained as Boolean covers of the unique tight regular polytope of the same type.

Finite 3-connected-set-homogeneous locally 2Kn graphs and s-arc-transitive graphs

In this paper, all graphs are assumed to be finite. For s≥1 and a graph Γ, if for every pair of isomorphic connected induced subgraphs on at most s vertices there exists an automorphism of Γ mapping the first to the second, then we say that Γ is s-connected-set-homogeneous, and if every isomorphism between two isomorphic connected induced subgraphs on at most s vertices can be extended to an automorphism of Γ, then we say that Γ is s-connected-homogeneous. For n≥1, a graph Γ is said to be locally 2Kn if the subgraph [Γ(u)] induced on the set of vertices of Γ adjacent to a given vertex u is isomorphic to 2Kn.Note that 2-connected-set-homogeneous but not 2-connected-homogeneous graphs are just the half-arc-transitive graphs which are a quite active topic in algebraic graph theory. Motivated by this, we posed the problem of characterizing or classifying 3-connected-set-homogeneous graphs of girth 3 which are not 3-connected-homogeneous in Zhou (2021) [54]. Until now, there have been only two known families of 3-connected-set-homogeneous graphs of girth 3 which are not 3-connected-homogeneous, and these graphs are locally 2Kn with n=2 or 4. In this paper, we complete the classification of finite 3-connected-set-homogeneous graphs which are locally 2Kn with n≥2, and all such graphs are line graphs of some specific 2-arc-transitive graphs. Furthermore, we give a good description of finite 3-connected-set-homogeneous but not 3-connected-homogeneous graphs which are locally 2Kn and have solvable automorphism groups. This is then used to construct some new 3-connected-set-homogeneous but not 3-connected-homogeneous graphs as well as some new 2-arc-transitive graphs.

Pentavalent Cayley graphs of order twice a prime square with solvable automorphism groups

A Cayley graph [Formula: see text] on a finite group [Formula: see text] is said to be normal if the right regular representation [Formula: see text] of [Formula: see text] is normal in the full automorphism group of [Formula: see text]. In this paper, we classify all connected pentavalent non-normal Cayley graphs of order [Formula: see text] with solvable automorphism groups and [Formula: see text] a prime.

Symmetric cubic graphs with non-solvable automorphism groups

A cubic graph Γ is called G-symmetric if a group G of automorphisms of Γ acts transitively on the arcs of Γ, and G-basic if it is G-symmetric and G has no non-trivial normal subgroups with more than two orbits on the vertex set of Γ. We say the graph Γ is basic if it is G-basic for all arc-transitive subgroups G of Aut(Γ). All basic symmetric cubic graphs with solvable automorphism groups have been classified in Feng et al. (2015) [12]. In this paper, a characterization of basic symmetric cubic graphs with non-solvable automorphism groups is given.

A new flag-transitive affine plane of order bm$2^9$

Finite flag-transitive affine planes have received much attention during the past fifty years because of their connections with other combinatorial objects such as spreads, planar functions, semifields and linearized polynomials. In 1964, Foulser completely determined the automorphism groups of finite flag-transitive affine planes. If a flag-transitive affine plane has a solvable automorphism group, then the affine plane is called solvable. The non-solvable flag-transitive affine planes have been completely classified in the 1990s. But the complete classification for the solvable case seems far out of reach. All known solvable flag-transitive affine planes can be classified into two types: $\\mathcal{C}$-planes and $\\mathcal{H}$-planes, where $\\mathcal{H}$-planes only occur in the odd characteristic case. In this paper, we construct the first flag-transitive affine plane of order $2^9$ over its kernel $\\bF_{2^3}$, which is not of type $\\mathcal{C}$ and the largest Singer subgroup of the translation complement has order $(2^3-1)(2^9+1)/9$.

On the solvability of graded Novikov algebras

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a [Formula: see text]-graded Novikov algebra [Formula: see text] over a field [Formula: see text] with solvable [Formula: see text]-component [Formula: see text] is solvable, where [Formula: see text] is a finite additive abelean group and the characteristic of [Formula: see text] does not divide the order of the group [Formula: see text]. We also show that any Novikov algebra [Formula: see text] with a finite solvable group of automorphisms [Formula: see text] is solvable if the algebra of invariants [Formula: see text] is solvable.

Two infinite families of chiral polytopes of type {4,4,4} with solvable automorphism groups

We construct two infinite families of locally toroidal chiral polytopes of type {4,4,4}, with 1024m2 and 2048m2 automorphisms for every positive integer m, respectively. The automorphism groups of these polytopes are solvable groups, and when m is a power of 2, they provide examples with automorphism groups of order 2n where n can be any integer greater than 9. (On the other hand, no chiral polytopes of type [4,4,4] exist for n≤9.) In particular, our two families give a partial answer to a problem proposed by Schulte and Weiss in [21].

On cubic symmetric non-Cayley graphs with solvable automorphism groups

It was proved in Feng et al. (2015) that a cubic symmetric graph with a solvable automorphism group is either a Cayley graph or a 2-regular graph of type 22, that is, a graph with no automorphism of order 2 interchanging two adjacent vertices. In this paper an infinite family of non-Cayley cubic 2-regular graphs of type 22 with a solvable automorphism group is constructed, and the smallest graph has order 6174. This answers a question posed by Estélyi and Pisanski in 2016. Moreover, it includes a subfamily of graphs which are connected 2-regular covers of the Pappus graph with covering transformation group Zp3, and these graphs were missed in Oh (2009).

English

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, all connected valency seven symmetric graphs of order 2pq are classified, where p, q are distinct primes. It follows from the classification that there is a unique connected valency seven symmetric graph of order 4p, and that for odd primes p and q, there is an infinite family of connected valency seven one-regular graphs of order 2pq with solvable automorphism groups, and there are four sporadic ones with nonsolvable automorphism groups, which is 1, 2, 3-arc transitive, respectively. In particular, one of the four sporadic ones is primitive, and the other two of the four sporadic ones are bi-primitive.

The geometry of the Artin-Schreier-Mumford curves over an algebraically closed field

For a power $q$ of a prime $p$, the Artin-Schreier-Mumford curve $ASM(q)$ of genus $g=(q-1)^2$ is the nonsingular model $\mathcal{X}$ of the irreducible plane curve with affine equation $(X^q+X)(Y^q+Y)=c,\, c\neq 0,$ defined over a field $\mathbb{K}$ of characteristic $p$. The Artin-Schreier-Mumford curves are known from the study of algebraic curves defined over a non-Archimedean valuated field since for $|c|<1$ they are curves with a large solvable automorphism group of order $2(q-1)q^2 =2\sqrt{g}(\sqrt{g}+1)^2$, far away from the Hurwitz bound $84(g-1)$ valid in zero characteristic. In this paper we deal with the case where $\mathbb{K}$ is an algebraically closed field of characteristic $p$. We prove that the group $Aut(\mathcal{X})$ of all automorphisms of $\mathcal{X}$ fixing $\mathbb{K}$ elementwise has order $2q^2(q-1)$ and it is the semidirect product $Q\rtimes D_{q-1}$ where $Q$ is an elementary abelian group of order $q^2$ and $D_{q-1}$ is a dihedral group of order $2(q-1)$. For the special case $q=p$, this result was proven by Valentini and Madan. Furthermore, we show that $ASM(q)$ has a nonsingular model $\mathcal{Y}$ in the three-dimensional projective space $PG(3,\mathbb{K})$ which is neither classical nor Frobenius classical over the finite field $\mathbb{F}_{q^2}$.

Semiregular automorphisms in vertex-transitive graphs with a solvable group of automorphisms

It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally 2-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. The known affirmative answers for graphs with primitive and quasiprimitive groups of automorphisms suggest that solvable groups need to be considered if one is to hope for a complete solution of this conjecture. It is the purpose of this paper to present an overview of known results and suggest possible further lines of research towards a complete solution of the problem.

Finite flag-transitive affine planes with a solvable automorphism group

In this paper, we consider finite flag-transitive affine planes with a solvable automorphism group. Under a mild number-theoretic condition involving the order and dimension of the plane, the translation complement must contain a linear cyclic subgroup that either is transitive or has two equal-sized orbits on the line at infinity. We develop a new approach to the study of such planes by associating them with planar functions and permutation polynomials in the odd order and even order case respectively. In the odd order case, we characterize the Kantor–Suetake family by using Menichetti's classification of generalized twisted fields and Blokhuis, Lavrauw and Ball's classification of rank two commutative semifields. In the even order case, we develop a technique to study permutation polynomials of DO type by quadratic forms and characterize such planes that have dimensions up to four over their kernels.

Tetravalent half-arc-transitive graphs of order a product of three primes

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let n be a product of three primes. The problem on the classification of the tetravalent half-arc-transitive graphs of order n has been considered by Xu (1992), Feng et al. (2007) and Wang and Feng (2010), and it was solved for the cases where n is a prime cube or twice a product of two primes. In this paper, we solve this problem for the remaining cases. In particular, there exist some families of these graphs which have a solvable automorphism group but are not metacirculants.

Nowhere-zero 3-flows in graphs admitting solvable arc-transitive groups of automorphisms

Tutte's 3-flow conjecture asserts that every 4-edge-connected graph has a nowhere-zero 3-flow. In this note we prove that every regular graph of valency at least four admitting a solvable arc-transitive group of automorphisms admits a nowhere-zero 3-flow.

$${C_2}$$ C 2 -Cofiniteness of Cyclic-Orbifold Models

We prove an orbifold conjecture for a solvable automorphism group. Namely, we show that if V is a C_2-cofinite simple vertex operator algebra and G is a finite solvable automorphism group of V, then the fixed point vertex operator subalgebra V^G is also C_2-cofinite. This offers a mathematically rigorous background to orbifold theories of finite type with solvable automorphism groups.

Normality of 2-Cayley digraphs

A digraph Γ is called a 2-Cayley digraph over a group G, if there exists a semiregular subgroup RG of Aut(Γ) isomorphic to G with two orbits. We say that Γ is normal if RG is a normal subgroup of Aut(Γ). In this paper, we determine the normalizer of RG in Aut(Γ). We show that the automorphism group of each normal 2-Cayley digraph over a group with solvable automorphism group, is solvable. We prove that for each finite group G≠Q8×Z2r, r≥0, where Q8 is the quaternion group of order 8 and Z2 is the cyclic group of order 2, there exists a normal 2-Cayley graph over G and that every finite group has a normal 2-Cayley digraph.

Symmetric cubic graphs with solvable automorphism groups

A cubic graph Γ is G-arc-transitive if G≤Aut(Γ) acts transitively on the set of arcs of Γ, and G-basic if Γ is G-arc-transitive and G has no non-trivial normal subgroup with more than two orbits. Let G be a solvable group. In this paper, we first classify all connected G-basic cubic graphs and determine the group structure for every G. Then, combining covering techniques, we prove that a connected cubic G-arc-transitive graph is either a Cayley graph, or its full automorphism group is of type 22, that is, a 2-regular group with no involution reversing an edge, and has a non-abelian normal subgroup such that the corresponding quotient graph is the complete bipartite graph of order 6.

Pentavalent symmetric graphs of order [formula omitted

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, all connected pentavalent symmetric graphs of order 2pq are classified, where p,q are distinct primes. It follows from the classification that there are two connected pentavalent symmetric graphs of order 4p, and that for odd primes p and q, there is an infinite family of connected pentavalent symmetric graphs of order 2pq with solvable automorphism groups and there are seven sporadic ones with nonsolvable automorphism groups.

An infinite family of half-arc-transitive graphs with universal reachability relation

The action of a subgroup G of automorphisms of a graph X is said to be half-arc-transitive if it is vertex-transitive and edge-transitive but not arc-transitive. In this case the graph X is said to be G - half-arc-transitive. In particular, when G = Aut X the graph X is said to be half-arc-transitive. Two oppositely oriented digraphs may be associated with any such graph in a natural way. The reachability relation of a graph admitting a half-arc-transitive group action is an equivalence relation defined on either of these two digraphs as follows. An arc e is reachable from an arc e ′ if there exists an alternating path starting with e and ending with e ′ . The reachability relation is clearly universal for all vertex-primitive half-arc-transitive graphs. The smallest known vertex-primitive half-arc-transitive graphs have valency 14 and no such graphs of valency smaller than 10 exist. The natural framework for the question of the existence of half-arc-transitive graphs with universal reachability relation is therefore the family of vertex-imprimitive half-arc-transitive graphs, and in particular those of valency less than 14. It is known that no such graph of valency 4 exists (see D. Marušič, Half-transitive group actions on finite graphs of valency 4, J. Combin. Theory Ser. B 73 (1998) 41–76). In this paper an infinite family of vertex-imprimitive half-arc-transitive graphs of valency 12 with universal reachability relation is constructed. These graphs have a solvable automorphism group but are not metacirculants.