Groups acting on tensor products

TL;DR

This paper provides a detailed analysis of groups preserving tensor products over certain rings, offering algorithms for their generation and applications to computational problems in algebra.

Contribution

It offers a comprehensive description of groups preserving tensor products over semisimple and semiprimary rings, including effective algorithms for their construction.

Findings

Explicit descriptions of groups preserving tensor products

Algorithms for generating these groups

Applications to computational algebra problems

Abstract

Groups preserving a distributive product are encountered often in algebra. Examples include automorphism groups of associative and nonassociative rings, classical groups, and automorphism groups of p-groups. While the great variety of such products precludes any realistic hope of describing the general structure of the groups that preserve them, it is reasonable to expect that insight may be gained from an examination of the universal distributive products: tensor products. We give a detailed description of the groups preserving tensor products over semisimple and semiprimary rings, and present effective algorithms to construct generators for these groups. We also discuss applications of our methods to algorithmic problems for which all currently known methods require an exponential amount of work.