J. I. Hall

Locally petersen graphs

A graph Γ is locally Petersen if, for each point t of Γ, the graph induced by Γ on all points adjacent to t is isomorphic to the Petersen graph. We prove that there are exactly three isomorphism classes of connected, locally Petersen graphs and further characterize these graphs by certain of their parameters.

Some Groups Generated by Transvection Subgroups

In the 1960’s, J. E. McLaughlin [lo, 111 determined all irreducible finite dimensional linear groups generated by transvection subgroups. (A trans- section of a k-vector space V is linear map t(cp, x): V-t given by vt(cp, x) = 2) + VqJ .Y for all t’ E V, where XE Vand (PE V* are non-zero and xcp = 0; V* is the dual space of V A k-transvection subgroup T(cp, x) consists of the identity and all r(cp’, x’), where q’ and x’ range over the l-dimensional subspaces spanned by cp and x, respectively; it is isomorphic to the additive group of k.) McLaughlin showed that the only such groups are the special linear and symplectic groups and, in the case when k is F, (the field with two elements), the orthogonal and symmetric groups. It is our purpose to give a new proof of a result which includes McLaughlin’s; we discard the assumptions of finiteness of the dimension of V over k and also (to some extent) the irreducibility of the group, while still obtaining a result which is recognizably like McLaughlin’s. Our result for irreducible groups is the following. (Unexplained terminol- ogy will be defined after the statement of the theorem; the more com- plicated result under a weaker hypothesis is stated near the end of the Introduction.)

Infinite Alternating Groups as Finitary Linear Transformation Groups

abstract degree of a classical group as its natural projective representation degree (see (6.2)). Case (1) of (2.6) is dispensed with using Mal’cev’s local characterization of finite dimensional linear groups and their classification. THEOREM (2.7) (Mal’cev, see [ 13, l.L.91). Let G be a Iocally finite group with adeg(G) < 00. Then G is a finite dimensional linear group. THEOREM (2.8) [l, 2,8,21]. Let G be an infinite group. Then is a simple, locally finite, linear group of finite degree if and only if G is isomorphic to an adjoint group of Lie type over an infinite subfield Fp, the algebraic closure of [F,, for some prime p. 3. LOCAL CHARACTERIZATION OF FINITARY LINEAR GROUPS; A THEOREM OF MAL’CEV TYPE Mal’cev’s theorem (2.7) provides a local characterization of locally finite linear groups G of finite degree. The property of having a faithful linear representation of degree n passes from the set of finite subgroups of G to G itself. Under this section we provide a similar result for locally finite simple groups which are linitary linear. Remember that the linear transformation g of the vector space I/ is finitary if g fixes pointwise a subspace of finite codimension in V. That is, the subspace C,(g) = {v E V 1 vg = v} has finite codimension in V, or equivalently the subspace [V, g] = {v(g - 1) 1 v E V} has finite dimension in V. The collection of all linitary members of GL( V) forms a normal sub- group FGL( V), and a group which is (isomorphic to) a subgroup of FGL( V), for some V, is sometimes called a finitary linear group.

Primitive axial algebras of Jordan type

Abstract An axial algebra over the field F is a commutative algebra generated by idempotents whose adjoint action has multiplicity-free minimal polynomial. For semisimple associative algebras this leads to sums of copies of F . Here we consider the first nonassociative case, where adjoint minimal polynomials divide ( x − 1 ) x ( x − η ) for fixed 0 ≠ η ≠ 1 . Jordan algebras arise when η = 1 2 , but our motivating examples are certain Griess algebras of vertex operator algebras and the related Majorana algebras. We study a class of algebras, including these, for which axial automorphisms like those defined by Miyamoto exist, and there classify the 2-generated examples. Always for η ≠ 1 2 and in identifiable cases for η = 1 2 this implies that the Miyamoto involutions are 3-transpositions, leading to a classification.

The hyperbolic lines of finite symplectic spaces

1. INTRODUCTION The classical theorem of Veblen and Young [14] characterizes the Desarguesian projective spaces of dimension at least three as those connec- ted partial linear spaces in which each pair of intersecting lines lies in a proper subspace which is a projective plane. In this article we present a similar result.

Graphs with constant link and small degree or order

The graph G has constant link L if for each vertex x of G the graph induced by G on the vertices adjacent to x is isomorphic to L. For each graph L on 6 or fewer vertices we decide whether or not there exists a graph G with constant link L. From this we are able to list all graphs on 11 or fewer vertices which have constant link.

Presentations of some 3-transposition groups

We classify all 3-transposition groups which are generated by at most five of their 3-transpositions. Modulo a center these are 27 specific groups plus various quotients of a particular group of order 2(349). For each of the 27 groups at least one presentation is given. We also give presentations for many groups of importance in the recent classification of 3-transposition groups with trivial center. Our presentations include ones for the three sporadic 3-transposition groups of Fischer, each on a 3-transposition generating set of minimal size.

A local characterization of the Johnson scheme

Within the Johnson schemeI(m, d) we find the graphK(m, d) ofd-subsets of anm-set, two such adjacent when disjoint. Among all connected graphs,K(m, d) is characterized by the isomorphism type of its vertex neighborhoods providedm is sufficiently large compared tod.

Equidistant codes with distance 12

Abstract Let C be an equidistant binary code with m words, pairwise at distance 2 k , It is known that if C is not trivial then m ⩽ k 2 + k + 2. Furthermore equality is possible if and only if a projective plane of order k exists. This settles the problem of determining the maximal m for k ⩽10 with the exception of k = 6. In this paper we show that if k = 6 then m ⩽ 32, and we give an example of a code with m = 32. To settle the next unknown case, i.e. k = 10, one would first have to know whether a projective plane of order 10 exist.

The trace operator and redundancy of Goppa codes

The dimension of subfield subcodes and in particular generalized Goppa codes is studied using the kernel of an associated trace map. Bounds on the dimension for a particular general class of binary Goppa codes are obtained. Deeper study of some special cases and other related polynomials yields still tighter bounds for any s>1. >