In this paper we find the largest automorphism group of a smooth cubic surface over any finite field of characteristic 2. We prove that if the order of the field is a power of 4, then the ...automorphism group of maximal order of a smooth cubic surface over this field is PSU4(F2). If the order of the field of characteristic 2 is not a power of 4, then we prove that the automorphism group of maximal order of a smooth cubic surface over this field is the symmetric group of degree 6. Moreover, we prove that smooth cubic surfaces with such properties are unique up to isomorphism.
For two unital Kirchberg algebras with finitely generated K-groups, we introduce a property, called reciprocality, which is proved to be closely related to the homotopy theory of Kirchberg algebras. ...We show the Spanier–Whitehead duality for bundles of separable nuclear UCT C*-algebras with finitely generated K-groups and conclude that two reciprocal Kirchberg algebras share the same structure of their bundles.
Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical ...argument of Quillen, homological stability for the family of groups. We show that stability also holds with both polynomial and abelian twisted coefficients, with no further assumptions. This new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups, such as the symmetric groups, general linear groups and mapping class groups. By making systematic the proofs of classical stability results, we show that they all hold with the same type of coefficient systems, obtaining in particular without any further work new stability theorems with twisted coefficients for the symmetric groups, braid groups, automorphisms of free groups, unitary groups, mapping class groups of non-orientable surfaces and mapping class groups of 3-manifolds. Our construction can also be applied to families of groups not considered before in the context of homological stability.
As a byproduct of our work, we construct the braided analogue of the category FI of finite sets and injections relevant to the present context, and define polynomiality for functors in the context of pre-braided monoidal categories.
Given a field K, we investigate which subgroups of the group AutAK2 of polynomial automorphisms of the plane are linear or not.
The results are contrasted. The group AutAK2 itself is nonlinear, ...except if K is finite, but it contains some large subgroups, of “codimension-five” or more, which are linear. This phenomenon is specific to dimension two: it is easy to prove that any natural “finite-codimensional” subgroup of AutAK3 is nonlinear, even for a finite field K.
When chK=0, we also look at a similar questions for f.g. subgroups, and the results are again disparate. The group AutAK2 has a one-related f.g. subgroup which is not linear. However, there is a large subgroup, of “codimension-three”, which is locally linear but not linear.
In two previous papers we constructed new families of completely regular codes by concatenation methods. Here we determine cases in which the new codes are completely transitive. For these cases we ...also find the automorphism groups of such codes. For the remaining cases, we show that the codes are not completely transitive assuming an upper bound on the order of the monomial automorphism groups, according to computational results.
In this paper, we observe the possibility that the group \(S_{n}\times S_{m}\) acts as a flag-transitive automorphism group of a block design with point set \(\{1,\ldots ,n\}\times \{1,\ldots ...,m\},4\leq n\leq m\leq 70\). We prove the equivalence of that problem to the existence of an appropriately defined smaller flag-transitive incidence structure. By developing and applying several algorithms for the construction of the latter structure, we manage to solve the existence problem for the desired designs with \(nm\) points in the given range. In the vast majority of the cases with confirmed existence, we obtain all possible structures up to isomorphism.
Locally repairable codes, or locally recoverable codes (LRC for short), are designed for applications in distributed and cloud storage systems. Similar to classical block codes, there is an important ...bound called the Singleton-type bound for locally repairable codes. In this paper, an optimal locally repairable code refers to a block code achieving this Singleton-type bound. Like classical MDS codes, optimal locally repairable codes carry some very nice combinatorial structures. Since the introduction of the Singleton-type bound for locally repairable codes, people have put tremendous effort into construction of optimal locally repairable codes. There are a few constructions of optimal locally repairable codes in the literature. Most of these constructions are realized via either combinatorial or algebraic structures. In this paper, we apply automorphism group of the rational function field to construct optimal locally repairable codes by considering the group action on projective lines over finite fields. Due to various subgroups of the projective general linear group, we are able to construct optimal locally repairable codes with flexible locality as well as smaller alphabet size comparable to the code length. In particular, we produce new families of <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula>-ary locally repairable codes, including codes of length <inline-formula> <tex-math notation="LaTeX">q+1 </tex-math></inline-formula> via cyclic groups.
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