Subalgebras of Octonion Algebras

For an arbitrary octonion algebra, we determine all subalgebras. It turns out that every subalgebra of dimension less than four is associative, while every subalgebra of dimension greater than four is not associative. In any split octonion algebra, there are both associative and non-associative subalgebras of dimension four. Except for one-dimensional subalgebras spanned by idempotents, any two isomorphic subalgebras are in the same orbit under automorphisms

Confluence Graphs of Unitals

We show that the cliques of maximal size in the confluence graph of an arbitrary unital of order $q>2$ have size $q^2$, and that these cliques are the pencils of all blocks through a given point. This solves the Erd\H{o}s-Ko-Rado problem for all unitals. We also determine all maximal cliques of the confluence graph of the Hermitian unitals. As an application, we show that the confluence graph of an arbitrary unital unambiguously determines the unital. Along the way, we show that each linear space with $q^2$ points such that the sizes of both point rows and line pencils are bounded above by $q+1$ embeds in a projective plane of order $q$

Embeddings of hermitian unitals into pappian projective planes

Every embedding of a hermitian unital with at least four points on a block into any pappian projective plane is standard, i.e. it originates from an inclusion of the pertinent fields. This result about embeddings also allows us to determine the full automorphism groups of (generalized) hermitian unitals

Moufang sets generated by translations in unitals

We consider unitals of order q with two points which are centers of translation groups of order q. The group G generated by these translations induces a Moufang set on the block joining the two points. We show that G is either SL ( 2 , F q ) (as in all classical unitals and also in some nonclassical examples), or PSL ( 2 , F q ), or a Suzuki, or a Ree group. Moreover, G is semiregular outside the special block

Linear spaces embedded into projective spaces via Baer sublines

Every nontrivial linear space embedded in a Pappian projective space such that the blocks of the linear space are projectively equivalent Baer sublines with respect to a separable quadratic field extension is either a Baer subspace, or a Hermitian unital.Mathematics Subject Classifications: 51A45, 51E1

Finite subunitals of the Hermitian unitals

Every finite subunital of any generalized hermitian unital is itself a hermitian unital; the embedding is given by an embedding of quadratic field extensions. In particular, a generalized hermitian unital with a finite subunital is a hermitian one (i.e., it originates from a separable field extension)

Embeddings of unitals such that each block is a subline

We consider unitals embedded in a pappian projective plane such that every block is a subline. We show that every such unital is a hermitian one, and that the embedding is standard