James B. Wilson

Finding central decompositions of p-groups

Abstract A Las Vegas polynomial-time algorithm is given to find a central decomposition of maximum size for a finite p-group of class 2. The proof introduces an associative *-ring as a tool for studying central products of p-groups. This technique leads to a translation of the problem into classical linear algebra which can be solved by application of the MeatAxe and other established module-theoretic algorithms. When p is small, our algorithm runs in deterministic polynomial time.

Computing isometry groups of Hermitian maps

A theorem is proved on the structure of the group of isometries of a Hermitian map b : V × V → W , where V and W are vector spaces over a finite field of odd order. Also a Las Vegas polynomial-time algorithm is presented which, given a Hermitian map, finds generators for, and determines the structure of its isometry group. The algorithm can be adapted to construct the intersection over a set of classical subgroups of GL(V ), giving rise to the first polynomial-time solution of this old problem. The approach yields new algorithmic tools for algebras with involution, which in turn have applications to other computational problems of interest. Implementations of the various algorithms in the Magma system demonstrate their practicability.

ISOMORPHISM IN EXPANDING FAMILIES OF INDISTINGUISHABLE GROUPS

Abstract. For every odd prime and every integer , there is a Heisenberg group of order that has pairwise nonisomorphic quotients of order . Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most . They are also directly and centrally indecomposable and of the same indecomposability type. Nevertheless, there is a polynomial-time algorithm to test for isomorphisms between these groups.

Existence, algorithms, and asymptotics of direct product decompositions, I

Abstract. Direct products of finite groups are a simple method to construct new groups from old ones. A difficult problem by comparison is to prove a generic group is indecomposable, or locate a proper nontrivial direct factor. To solve this problem it is shown that in most circumstances has a proper nontrivial subgroup such that every maximal direct product decomposition of induces a unique set of subgroups of where and for each , the nonabelian direct factors of are direct factors of . In particular, is indecomposable if and is contained in the Frattini subgroup of . This “local-global” property of direct products can be applied inductively to and so that the existence of a proper nontrivial direct factor depends on the direct product decompositions of the chief factors of . Chief factors are characteristically simple groups and therefore a direct product of isomorphic simple groups. Thus a search for proper direct factors of a group of size is reduced from the global search through all normal subgroups to a search of local instances induced from chief factors. There is one family of groups where no subgroup admits the local-global property just described. These are -groups of nilpotence class 2. There are isomorphism types of class 2 groups with order , which prevents a case-by-case study. Also these groups arise in the course of the induction described above so they cannot be ignored. To identify direct factors for nilpotent groups of class 2, a functor is introduced to the category of commutative rings. The result being that indecomposable -groups of class 2 are identified with local commutative rings. This relationship has little to do with the typical use of Lie algebras for -groups and is one of the essential and unexpected components of this study. These results are the by-product of an efficient polynomial-time algorithm to prove indecomposability or locate a proper nontrivial direct factor. The theorems also explain how many isomorphism types of indecomposable groups exists of a given order and how many direct factors a group can have. These two topics are explained in a second part to this paper.A polynomial-time algorithm is produced which, given generators for a group of permutations on a finite set, returns a direct product decomposition of the group into directly indecomposable subgroups. The process uses bilinear maps and commutative rings to characterize direct products of pgroups of class 2 and reduces general groups to p-groups using group varieties. The methods apply to quotients of permutation groups and operator groups as well.

A fast isomorphism test for groups whose Lie algebra has genus 2

Motivated by the need for efficient isomorphism tests for finite groups, we present a polynomial-time method for deciding isomorphism within a class of groups that is well-suited to studying local properties of general finite groups. We also report on the performance of an implementation of the algorithm in the computer algebra system {\sc magma}.

Division, Adjoints, and Dualities of Bilinear Maps

The distributive property can be studied through bilinear maps and various morphisms between these maps. The adjoint-morphisms between bilinear maps establish a complete abelian category with projectives and admits a duality. Thus the adjoint category is not a module category but nevertheless it is suitably familiar. The universal properties have geometric perspectives. For example, products are orthogonal sums. The nonsingular bilinear maps are the simple bimaps with respect to nondegenerate adjoint-morphisms. That formalizes the understanding that the atoms of linear geometries are algebraic objects with no zero-divisors. Adjoint-isomorphism coincides with principal isotopism; hence, semifields can be studied within this framework.

Optimal algorithms of Gram–Schmidt type

Three algorithms of Gram-Schmidt type are given that produce an orthogonal decomposition of finite

On automorphisms of groups, rings, and algebras

d

Decomposing p-groups via Jordan algebras

-dimensional symmetric, alternating, or Hermitian forms over division rings. The first uses

More characteristic subgroups, Lie rings, and isomorphism tests for p-groups

d^3/3+O(d^2)