On the Determination of the Maximum Order of the Group of a Tournament

1973 ◽  
Vol 16 (1) ◽  
pp. 11-14 ◽  
Author(s):  
B. Alspach ◽  
J. L. Berggren
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Maximum Order ◽  
Odd Order

Let denote the automorphism group of the tournament T. Let g(n) be the maximum of taken over all tournaments of order n. It was noted in [3] that g(n) is also the order of the subgroups of Sn of maximum odd order where Sn denotes the symmetric group of degree n.

A Combinatorial Proof of a Conjecture of Goldberg and Moon

1968 ◽  
Vol 11 (5) ◽  
pp. 655-661 ◽  
Author(s):  
Brian Alspach
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Combinatorial Proof ◽  
Odd Order ◽  
Equality Holding

Let Tn denote a tournament of order n, let G(Tn) denote the automorphism group of Tn, let |G| denote the order of the group G, and let g(n) denote the maximum of |G(Tn)| taken over all tournaments Tn of order n. Goldberg and Moon conjectured [2] that for all n≥1 with equality holding if and only if n is a power of 3. In an addendum to [2] it was pointed out that their conjecture is equivalent to the conjecture that if G is any odd order subgroup of Sn, the symmetric group of degree n, then with equality possible if and only if n is a power of 3. The latter conjecture was proved in [1] by John D. Dixon who made use of the Feit-Thompson theorem in his proof. In this paper we avoid use of the Feit-Thompson result and give a combinatorial proof of the Goldberg-Moon conjecture.

scholarly journals Analysis of Some Properties from Construction of Finite Projective Planes of Order 2 and 3

CAUCHY ◽  
2016 ◽  
Vol 4 (3) ◽  
pp. 131
Author(s):  
Vira Hari Krisnawati ◽  
Corina Karim
Keyword(s):  
Automorphism Group ◽  
Projective Plane ◽  
Symmetric Group ◽  
Block Design ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Steiner System ◽  
Incidence Relation ◽  
Finite Projective Planes ◽  
Set Of Points

<p class="abstract"><span lang="IN">In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system <em>S</em>(<em>t</em>, <em>k</em>, <em>v</em>) is a set of <em>v</em> points and <em>k</em> blocks which satisfy that every <em>t</em>-subset of <em>v</em>-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with <em>t</em> = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.</span></p><p class="abstract"><span lang="IN">In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.</span></p>

scholarly journals Varieties of nilpotent groups of class four (III)

1983 ◽  
Vol 35 (1) ◽  
pp. 109-122
Author(s):  
Patrick Fitzpatrick
Keyword(s):  
Nonnegative Integer ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Odd Order

AbstractIn this paper we complete the investigation of those varieties of nilpotent groups of class (at most) four whose free groups have no nontrivial elements of odd order. Each such variety is labelled by a vector of sixteen parameters, each parameter a nonnegative integer or ∞, subject to numerous but simple conditions. Each vector satisfying these conditions is in fact used and directly yields a defining set of laws for the variety it labels. Moreover, one can easily recognise from the parameters whether one variety is contained in another. In view of the reduction carried out in the first paper of this series (written jointly with L. G. Kovács) this completes the determination of all varieties of nilpotent groups of class four.

The Maximum Order of the Group of a Tournament

1967 ◽  
Vol 10 (4) ◽  
pp. 503-505 ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Automorphism Group ◽  
Maximum Order ◽  
Image Position

To each tournament Tn with n nodes n there corresponds the automorphism group G(Tn) consisting n of all dominance preserving permutations of the set of nodes. In a recent paper [3], Myron Goldberg and J. W. Moon consider the maximum order g(n) which the group of a tournament with n nodes may have. Among other results they prove that12

scholarly journals Varieties of nilpotent groups of class four (II)

1983 ◽  
Vol 35 (1) ◽  
pp. 74-108 ◽  
Author(s):  
Patrick Fitzpatrick
Keyword(s):  
Nonnegative Integer ◽  
Additional Assumption ◽  
Free Groups ◽  
Nilpotent Groups ◽  
The Third ◽  
Odd Order

AbstractThe first paper (written jointly with L. G. Kovács) of this three-part series reduced the problem of determining all varieties of the title to the study of the varieties of nilpotent groups of class (at most) four whose free groups have no nontrivial elements of odd order. The present paper deals with these under the additional assumption that the variety contains all nilpotent groups of class three. We label each such variety by a vector of eleven parameters, each parameter a nonnegative integer or ∞, subject to numerous but simple conditions. Each vector satisfying these conditions is in fact used, and matches directly a (finite) defining set of laws for the variety it labels. Moreover, one can readily recognize from the parameters whether one variety is contained in another. The third paper will complete the determination of all varieties of nilpotent groups of class four.

scholarly journals Translation planes of odd order and odd dimension

1979 ◽  
Vol 2 (2) ◽  
pp. 187-208 ◽  
Author(s):  
T. G. Ostrom
Keyword(s):  
Vector Space ◽  
Finite Groups ◽  
Translation Planes ◽  
Solvable Subgroup ◽  
Joint Paper ◽  
Desarguesian Planes ◽  
Odd Order ◽  
Collineation Groups ◽  
Zero Vector

The author considers one of the main problems in finite translation planes to be the identification of the abstract groups which can act as collineation groups and how those groups can act.The paper is concerned with the case where the plane is defined on a vector space of dimension2d overGF(q), whereqanddare odd. If the stabilizer of the zero vector is non-solvable, letG0be a minimal normal non-solvable subgroup. We suspect thatG0must be isomorphic to someSL(2,u)or homomorphic toA6orA7. Our main result is that this is the case whendis the product of distinct primes.The results depend heavily on the Gorenstein-Walter determination of finite groups having dihedral Sylow2-groups whendandqare both odd. The methods and results overlap those in a joint paper by Kallaher and the author which is to appear in Geometriae Dedicata. The only known example (besides Desarguesian planes) is Hering's plane of order27(i.e.,dandqare both equal to3) which admitsSL(2,13).

CONNECTIVITY AND SPECTRUM IN A GRAPH WITH A REGULAR AUTOMORPHISM GROUP OF ODD ORDER

1994 ◽  
Vol 04 (04) ◽  
pp. 529-560 ◽  
Author(s):  
JON A. SJOGREN
Keyword(s):  
Automorphism Group ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Laplacian Spectrum ◽  
Regular Automorphism ◽  
Fixed Integer ◽  
Quotient Graph ◽  
Odd Order ◽  
Induced Graph ◽  
Group Matrices

Let a finite group G of odd order n act regularly on a connected (multi-)graph Γ. That is, no group element other than the identity fixes any vertex. Then the “quotient graph” Δ under the action is the induced graph of orbits. We give a result about the connectivity of Γ and Δ in terms of their numbers of labeled spanning trees. In words, the spanning tree count of the graph is equal to n, the order of the given regular automorphism group, times the spanning tree count of the graph of orbits, times a perfect square integer. There is a dual result on the Laplacian spectrum saying that the multiset of Laplacian eigenvalues for the main graph is the disjoint union of the multiset for the quotient graph together with a multiset all of whose elements have even multiplicity. Specializing to the case of one orbit, we observe that a Cayley graph of odd order has spanning tree count equal to n times a square, and that that the Laplacian spectrum consists of the value 0 together with other doubled eigenvalues. These results are based on a study of matrices (and determinants) that consist of blocks of group-matrices. The generic determinant for such a matrix with the additional property of symmetry will have a dominanting square factor in its (multinomial) factorization. To show this, we make use of the Feit-Thompson theorem which provides a normal tower for an odd-order group, and perform a similarity conjugation with a fixed integer, unimodal matrix. Additional related results are given for certain group-matrices “twisted” by a group of automorphisms, generalizing the “g-circulants” of P.J. Davis.

scholarly journals $${\mathfrak {S}}_5$$-equivariant syzygies for the Del Pezzo surface of degree 5

2020 ◽  
Author(s):  
Ingrid Bauer ◽  
Fabrizio Catanese
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Blow Up ◽  
Pezzo Surface ◽  
Irreducible Representations ◽  
Del Pezzo Surface ◽  
Del Pezzo ◽  
Arithmetically Gorenstein ◽  
Pfaffian Equations

Abstract The Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in $${\mathbb {P}}^5$$P5 by the system of cubics passing through these points. It is the simplest example of the Buchsbaum–Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $${\mathfrak {S}}_5$$S5. We give canonical explicit $${\mathfrak {S}}_5$$S5-invariant Pfaffian equations through a 6$$\times $$×6 antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of $${\mathfrak {S}}_5$$S5. Finally, we give $${\mathfrak {S}}_5$$S5-invariant equations for the embedding of Y inside $$({\mathbb {P}}^1)^5$$(P1)5, and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4.

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