John Bamberg

The last two decades, since the seminal work of Selig [18], has seen projective geometric algebra (PGA) gain popularity as a modern coordinate-free framework for doing classical Euclidean geometry and other Cayley-Klein geometries. This framework is based upon a degenerate Clifford algebra, and it is the purpose of this paper to delve deeper into its internal algebraic structure and extract meaningful information for the purposes of PGA. This includes exploiting the split extension structure to realise the natural decomposition of elements of this Clifford algebra into Euclidean and ideal parts. This leads to a beautiful demonstration of how Playfair’s axiom for affine geometry arises from the ambient degenerate quadratic space. The highlighted split extension property of the Clifford algebra also corresponds to a splitting of the group of units and the Lie algebra of bivectors. Central to these results is that the degenerate Clifford algebra $${{\,\textrm{Cl}\,}}(V)$$ Cl ( V ) is isomorphic to the twisted trivial extension $${{\,\textrm{Cl}\,}}(V/\mathbb {F}{e_{0}})\ltimes _\alpha {{\,\textrm{Cl}\,}}(V/\mathbb {F}{e_{0}})$$ Cl ( V / F e 0 ) ⋉ α Cl ( V / F e 0 ) , where $${e_{0}}$$ e 0 is a degenerate vector and $$\alpha $$ α is the grade-involution.

Suppose we have a finite thick generalised quadrangle whose automorphism group $G$ acts primitively on both the set of points and the set of lines. Then $G$ must be almost simple. In this paper, we show that $\operatorname{soc}(G)$ cannot be isomorphic to $\operatorname{Sz}(2^{2m+1})$ or $\operatorname{Ree}(3^{2m+1})$ where $m$ is a positive integer.

We describe the spreading property for finite transitive permutation groups in terms of properties of their associated coherent configurations, in much the same way that separating and synchronising groups can be described via properties of their orbital graphs. We also show how the other properties in the synchronisation hierarchy naturally fit inside this framework. This combinatorial description allows for more efficient computational tools, and we deduce that every spreading permutation group of degree at most $8191$ is a $\mathbb{Q}$I-group. We also consider design-orthogonality more generally for noncommutative homogeneous coherent configurations.

The synchronization hierarchy of finite permutation groups consists of classes of groups lying between $2$ -transitive groups and primitive groups. This includes the class of spreading groups, which are defined in terms of sets and multisets of permuted points, and which are known to be primitive of almost simple, affine or diagonal type. In this paper, we prove that in fact no spreading group of diagonal type exists. As part of our proof, we show that all non-abelian finite simple groups, other than six sporadic groups, have a transitive action in which a proper normal subgroup of a point stabilizer is supplemented by all corresponding two-point stabilizers.

Let $${\mathcal {C}}$$C be a 430-cap of $$\textrm{PG}(6,4)$$PG ( 6 , 4 ) having two intersection sizes with respect to hyperplanes. We show that no hyperplane of $$\textrm{PG}(6,4)$$PG ( 6 , 4 ) intersects $${\mathcal {C}}$$C in a Hill 78-cap. So if it can be shown that the Hill 78-cap of $$\textrm{PG}(5,4)$$PG ( 5 , 4 ) is projectively unique, then such a 430-cap does not exist, or equivalently, a two-weight $$[430,7]_{\mathbb {F}_4}$$[ 430 , 7 ] F 4 linear code with dual weight at least 4, does not exist.

An affine vector space partition of $\operatorname{AG}(n,q)$ is a set of proper affine subspaces that partitions the set of points. Here we determine minimum sizes and enumerate equivalence classes of affine vector space partitions for small parameters. We also give parametric constructions for arbitrary field sizes.

An affine vector space partition of $${{\,\textrm{AG}\,}}(n,q)$$ AG ( n , q ) is a set of proper affine subspaces that partitions the set of points. Here we determine minimum sizes and enumerate equivalence classes of affine vector space partitions for small parameters. We also give parametric constructions for arbitrary field sizes.