This contains a new version of the so-called non-commutative Gauss algorithm for polycyclic groups. Its results allow to read off the order and the index of a subgroup in an (possibly infinite) polycyclic group.
The groups whose orders factorise into at most four primes have been described (up to isomorphism) in various papers. Given such an order n, this paper exhibits a new explicit and compact determination of the isomorphism types of the groups of order
n together with effective algorithms to enumerate, construct, and identify these groups. The algorithms are implemented for the computer algebra system GAP.
We present a new algorithm that, given two matrices in $GL(n,Q)$, decides if they are conjugate in $GL(n,Z)$ and, if so, determines a conjugating matrix. We also give an algorithm to construct a generating set for the centraliser in $GL(n,Z)$ of a ma
trix in $GL(n,Q)$. We do this by reducing these problems respectively to the isomorphism and automorphism group problems for certain modules over rings of the form $mathcal O_K[y]/(y^l)$, where $mathcal O_K$ is the maximal order of an algebraic number field and $l in N$, and then provide algorithms to solve the latter. The algorithms are practical and our implementations are publicly available in Magma.
We describe an algorithm to compute the Schur multipliers of all nilpotent Lie $p$-rings in the family defined by a symbolic nilpotent Lie $p$-ring. Symbolic nilpotent Lie $p$-rings can be used to describe the isomorphism types of $p$-groups of order
$p^n$ for $n leq 7$ and all primes $p geq n$. We apply our algorithm to compute the Schur multipliers of all $p$-groups of order dividing $p^6$.
A famous result of Hall asserts that the multiplication and exponentiation in finitely generated torsion free nilpotent groups can be described by rational polynomials. We describe an algorithm to determine such polynomials for all torsion free nilpo
tent groups of given Hirsch length. We apply this to determine the Hall polynomials for all such groups of Hirsch length at most 7.
Let $N(n)$ denote the number of isomorphism types of groups of order $n$. We consider the integers $n$ that are products of at most $4$ not necessarily distinct primes and exhibit formulas for $N(n)$ for such $n$.
A theorem by Hall asserts that the multiplication in torsion free nilpotent groups of finite Hirsch length can be facilitated by polynomials. In this note we exhibit explicit Hall polynomials for the torsion free nilpotent groups of Hirsch length at most 5.
Let $n$ be a positive integer and let $f_1, ldots, f_r$ be polynomials in $n^2$ indeterminates over an algebraically closed field $K$. We describe an algorithm to decide if the invertible matrices contained in the variety of $f_1, ldots, f_r$ form a
subgroup of $GL(n,K)$; that is, we show how to decide if the polynomials $f_1, ldots, f_r$ define a linear algebraic group.
We investigate the graded Lie algebras of Cartan type $W$, $S$ and $H$ in characteristic 2 and determine their simple constituents and some exceptional isomorphisms between them. We also consider the graded Lie algebras of Cartan type $K$ in characte
ristic 2 and conjecture that their simple constituents are isomorphic to Lie algebras of type $H$.
We introduce the concept of an infinite cochain sequence and initiate a theory of homological algebra for them. We show how these sequences simplify and improve the construction of infinite coclass families (as introduced by Eick and Leedham-Green) a
nd how they apply in proving that almost all groups in such a family have equivalent Quillen categories. We also include some examples of infinite families of p-groups from different coclass families that have equivalent Quillen categories.
