On groups with a class-preserving outer automorphism

In 1911, Burnside asked whether or not there exist groups that have an outer automorphism which preserves conjugacy classes. Two years later he answered his own question by constructing a family of such groups. Using the small group library in MAGMA we determine all of the groups of order n < 512 that possess such an automorphism. Our investigations led to the discovery of four new infinite families of such groups, all of which are 2-groups of coclass 4

Groups Acting on Tensor Products

Groups preserving a distributive product are encountered often in mathematics. Examples include automorphism groups of associative and non associative rings, classical groups, and automorphisms 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 such tensor products over semisimple and semi primary 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

On Groups with a Class-Preserving Outer Automorphism

Four infinite families of 2-groups are presented, all of whose members possess an outer automorphism that preserves conjugacy classes. The groups in these families are central extensions of their predecessors by a cyclic group of order 2. For each integer r\u3e1, there is precisely one 2-group of nilpotency class r in each of the four families. All other known families of 2-groups possessing a class-preserving outer automorphism consist entirely of groups of nilpotency class 2

On the ranks of string C-group representations for symplectic and orthogonal groups

We determine the ranks of string C-group representations of 4-dimensional projective symplectic groups over a finite field, and comment on those of higher-dimensional symplectic and orthogonal groups

Exact sequences of inner automorphisms of tensors

We produce a long exact sequence whose terms are unit groups of associative algebras that behave as inner automorphisms of a given tensor. Our sequence generalizes known sequences for associative and non-associative algebras. In a manner similar to those, our sequence facilitates inductive reasoning about, and calculation of the groups of symmetries of a tensor. The new insights these methods afford can be applied to problems ranging from understanding algebraic structures to distinguishing entangled states in particle physics

The Module Isomorphism Problem Reconsidered

Algorithms to decide isomorphism of modules have been honed continually over the last 30 years, and their range of applicability has been extended to include modules over a wide range of rings. Highly efficient computer implementations of these algorithms form the bedrock of systems such as GAP and MAGMA, at least in regard to computations with groups and algebras. By contrast, the fundamental problem of testing for isomorphism between other types of algebraic structures -- such as groups, and almost any type of algebra -- seems today as intractable as ever. What explains the vastly different complexity status of the module isomorphism problem? This paper argues that the apparent discrepancy is explained by nomenclature. Current algorithms to solve module isomorphism, while efficient and immensely useful, are actually solving a highly constrained version of the problem. We report that module isomorphism in its general form is as hard as algebra isomorphism and graph isomorphism, both well-studied problems that are widely regarded as difficult. On a more positive note, for cyclic rings we describe a polynomial-time algorithm for the general module isomorphism problem. We also report on a MAGMA implementation of our algorithm

Testing isomorphism of graded algebras

We present a new algorithm to decide isomorphism between finite graded algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it runs in time polynomial in the order of the input algebras. We introduce heuristics that often dramatically improve the performance of the algorithm and report on an implementation in Magma