Black Box White Arrow

The present paper proposes a new and systematic approach to the so-called black box group methods in computational group theory. Instead of a single black box, we consider categories of black boxes and their morphisms. This makes new classes of black box problems accessible. For example, we can enri...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Borovik, Alexandre, \c{S}ükrü Yalçinkaya
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 05.05.2014
Subjects
Automorphisms
Automotive parts
Axioms
Black boxes
Fields (mathematics)
Group theory
Shades
Online AccessGet full text

Cover

More Information
Summary:The present paper proposes a new and systematic approach to the so-called black box group methods in computational group theory. Instead of a single black box, we consider categories of black boxes and their morphisms. This makes new classes of black box problems accessible. For example, we can enrich black box groups by actions of outer automorphisms. As an example of application of this technique, we construct Frobenius maps on black box groups of untwisted Lie type in odd characteristic (Section 6) and inverse-transpose automorphisms on black box groups encrypting \({\rm (P)SL}_n(\mathbb{F}_q)\). One of the advantages of our approach is that it allows us to work in black box groups over finite fields of big characteristic. Another advantage is explanatory power of our methods; as an example, we explain Kantor's and Kassabov's construction of an involution in black box groups encrypting \({\rm SL}_2(2^n)\). Due to the nature of our work we also have to discuss a few methodological issues of the black box group theory. The paper is further development of our text "Fifty shades of black" [arXiv:1308.2487], and repeats parts of it, but under a weaker axioms for black box groups.
ISSN:2331-8422