Order Of Automorphism Group Research Articles

Upper bounds of orders of automorphism groups of leafless metric graphs

We prove a tropical analogue of the theorem of Hurwitz: A leafless metric graph of genus g ≥ 2 has at most 12 automorphisms when g = 2 and 2 g g ! automorphisms when g ≥ 3 . These inequalities are optimal; for each genus, we give all metric graphs which have the maximum numbers of automorphisms. The proof is written in terms of graph theory.

Toward the complete set of fulleroids with four-membered rings: Predicted topologies and symmetry statistics for Cn (n = 20–60)

The fullerene-like structures can be characterized not only by the classical topology (12 pentagonal and rest hexagonal rings), but can also contain 4-membered rings (typically forbidden due to strong steric stresses). Here we report the numbers and the symmetry point group statistics for all C 20–C 60 fulleroids with 4-membered rings (141325 in total). The most symmetrical structures with 6 to 120 automorphism group orders (251 in total) are drawn in Schlegel diagrams and characterized by facet sets and symmetry point groups in the list. Among them, 19 series of tubulen-like structures constructed from pairs of the same caps separated by belts of hexagons have been found. The obtained results are of practical interest and can be used for the description of the topologies of polyoxometallates with the fullerene-related structures.

On the real combinations of cubes and octahedra.

All the real combinations of cubes and octahedra (77657 in total) are enumerated and characterized by facet symbols and symmetry point groups. The most symmetrical polyhedra (with automorphism group orders not less than 6, 163 in total) are shown. It is assumed that they represent the most probable forms of natural diamond crystals. The results are discussed with respect to the Curie dissymmetry principle.

New extremal self-dual binary codes of length 68 via composite construction, 𝔽<SUB align="right">2 + <i>u</i>𝔽<SUB align="right">2 lifts, extensions and neighbours

We describe a composite construction from group rings where the groups have orders 16 and 8. This construction is then applied to find the extremal binary self-dual codes with parameters [32, 16, 8] or [32, 16, 6]. We also extend this composite construction by expanding the search field which enables us to find more extremal binary self-dual codes with the above parameters and with different orders of automorphism groups. These codes are then lifted to 𝔽2 + u𝔽2, to obtain extremal binary images of codes of length 64. Finally, we use the extension method and neighbour construction to obtain new extremal binary self-dual codes of length 68. As a result, we obtain 28 new codes of length 68 which were not known in the literature before.

Automorphism groups of smooth plane curves

The author classifies finite groups acting on smooth plane curves of degree at least four. Furthermore, he gives an upper bound for the order of automorphism groups of smooth plane curves and determines the exceptional cases in terms of defining equations. This paper also contains a simple proof of the uniqueness of smooth plane curves with the full automorphism group of maximum order for each degree.

Finitep-groups of maximal class with ‘large’ automorphism groups

AbstractThe classification ofp-groups of maximal class still is a wide open problem. Coclass Conjecture W proposes a way to approach such a classification: It suggests that the coclass graph𝒢${\mathcal{G}}$associated with thep-groups of maximal class can be determined from a finite subgraph using certain periodic patterns. Here we consider the subgraph𝒢∗${{\mathcal{G}}^{\ast}}$of𝒢${\mathcal{G}}$associated with thosep-groups of maximal class whose automorphism group orders are divisible byp-1${p-1}$. We describe the broad structure of𝒢∗${{\mathcal{G}}^{\ast}}$by determining its so-called skeleton. We investigate the smallest interesting casep=7${p=7}$in more detail using computational tools, and propose an explicit version of Conjecture W for𝒢∗${{\mathcal{G}}^{\ast}}$for arbitraryp≥7${p\geq 7}$. Our results are the first explicit evidence in support of Conjecture W for a coclass graph of infinite width.

On the automorphism group of polyhedral graphs

A (4,6)-fullerene is a three connected cubic planar graph whose faces are squares and hexagons. In this paper, for a given (4,6)-fullerene graph F, we compute the order of automorphism group F. We also study some spectral properties of fullerene graphs via their automorphism group.

On the automorphisms of direct product of monogenic semigroups and monoids

This paper investigates the automorphism group of monogenic [4] semigroups (or monoids) to find its relationship with the automorphism group of cyclic groups. Then, by considering a presentation related to the direct product of monogenic semigroups, verify the relationship between the automorphism group of These and the automorphism group of the group presented by the same presentation. This study gives us some [4]explicit formulas for computing the order of automorphism groups of these algebraic structures.

Intersection numbers and automorphisms of stable curves

Due to the orbifold singularities, the intersection numbers on the moduli space of curves $\bar{\sM}_{g,n}$ are in general rational numbers rather than integers. We study the properties of the denominators of these intersection numbers and their relationship with the orders of automorphism groups of stable curves. In particular, we prove a conjecture of Itzykson and Zuber. We also present a conjectural multinomial type numerical property for Hodge integrals.

Catalog of dessins d’enfants with no more than 4 edges

In this work all the dessins d'enfant with no more than 4 edges are listed and their Belyi pairs are computed. In order to enumerate all dessins the technique of matrix model computations was used. The total number of dessins is 134; among them 77 are spherical, 53 of genus 1 and 4 of genus 2. The orders of automorphism groups of all the dessins are also found. Dessins are listed by the number of edges. Dessins with the same number of edges are ordered lexicographically by their lists of 0-valencies. The corresponding matrix model for any list of 0-valencies is given and computed. Complex matrix models for dessins with 1 -- 3 edges are used. For the dessins with 4 edges we use Hermitian matrix model, correlators for which are computed in [1].

On enumeration of nonequivalent perfect binary codes of length 15 and rank 15

All nonequivalent perfect binary codes of length 15 and rank 15 are constructed that are obtained from the Hamming code H 15 by translating its disjoint components. Also, the main invariants of this class of codes are determined such as the ranks, dimensions of kernels, and orders of automorphism groups.

On the binary projective codes with dimension 6

All binary projective codes of dimension up to 6 are classified. Information about the number of the codes with different minimum distances and automorphism group orders is given.

On the symmetry of simple 16-hedra

The symmetry point group statistics for all combinatorially non-isomorphic convex simple 16-hedra (17490241 in total) are contributed in the paper for the first time. The most symmetrical polyhedra with 6 to 56 automorphism group orders (165 in total) are drawn in Schlegel diagrams and characterized by facet symbols and symmetry point groups.

Automorphism groups of 2-groups

A well-known conjecture on p-groups states that every non-abelian p-group G has the property that | G | divides | Aut ( G ) | . We exhibit periodic patterns in the automorphism group orders of the 2-groups of fixed coclass and we use this to show that for every positive integer r there are at most finitely many counterexamples to the conjecture among the 2-groups of coclass r.

Vasil’ev codes of length n = 2m and doubling of Steiner systems S(n, 4, 3) of a given rank

Extended binary perfect nonlinear Vasil'ev codes of length n = 2m and Steiner systems S(n, 4, 3) of rank n-m over F 2 are studied. The generalized concatenated construction of Vasil'ev codes induces a variant of the doubling construction for Steiner systems S(n, 4, 3) of an arbitrary rank r over F 2. We prove that any Steiner system S(n = 2m, 4, 3) of rank n-m can be obtained by this doubling construction and is formed by codewords of weight 4 of these Vasil'ev codes. The length 16 is studied in detail. Orders of the full automorphism groups of all 12 nonequivalent Vasil'ev codes of length 16 are found. There are exactly 15 nonisomorphic systems S(16, 4, 3) of rank 12 over F 2, and they can be obtained from codewords of weight 4 of the extended Vasil'ev codes. Orders of the automorphism groups of all these Steiner systems are found.

On the orders of automorphism groups of finite groups. II

In the Kourovka Notebook, Deaconescu asks if |AutG| > φ(|G|) for all finite groups G, where φ denotes the Euler totient function; and whether G is cyclic whenever |AutG| = φ(|G|). In an earlier paper we have answered both questions in the negative, and shown that |AutG|/φ(|G|) can be made arbitrarily small. Here we show that these results remain true if G is restricted to being perfect, or soluble. 1. The question, and general overview Let φ denote the Euler totient function, so that φ(n) is the number of integers m with 1 6 m 6 n such that m and n are coprime, and φ(n) n = r ∏

The variety of convex 12-hedra revised

The symmetry point group statistics for all combinatorially non-isomorphic convex 12-hedra (6384634 in total) are contributed in the paper. The numbers of 12-hedra with 11 to 19 vertices are revised. The most symmetrical shapes with 6 to 120 automorphism group orders (115 in total) are drawn in Schlegel diagrams and characterized by facet symbols and symmetry point groups.

On symmetry properties of molecules

It is well known to associate an Euclidean graph to a molecule. Balasubramanian computed the Euclidean graphs and their automorphism groups for benzene, eclipsed and staggered forms of ethane and eclipsed and staggered forms of ferrocene [see Chem. Phys. Lett. 232 (1995), 415]. In this Letter, we proved an algorithm, which is useful for finding symmetry of molecules. Using this algorithm, a new simple method is described, by means of which it is possible to calculate the automorphism group of weighted graphs. We apply this method to compute the symmetry of the smallest fullerene C 20. Finally, an upper bound for the order of automorphism group of weighted graphs is obtained.

On the symmetry of simple 14- and 15-hedra.

The symmetry point-group statistics for all combinatorially non-isomorphic simple 14- (339722) and 15-hedra (2406841 in total) are contributed in the paper for the first time. The most symmetrical shapes with 6 to 48 and 6 to 52 automorphism group orders (61 and 36, respectively) are drawn in Schlegel projections and characterized by facet symbols and symmetry point groups.

C72 to C100 fullerenes: combinatorial types and symmetries.

The symmetry point-group statistics for all combinatorially different C(72) to C(100) fullerenes with free 5-gonal facets only are suggested in the paper for the first time. The shapes with automorphism group orders not less than 3 are drawn in Schlegel diagrams.