This package contains various algorithms related to finite dimensional nilpotent associative algebras. We first give a brief introduction to these algebras and then an overview of the main algorithms.

Associative algebras and nilpotency

Let `A` be an associative algebra of dimension `d` over a field `F`.
Let `{b _{1}, ..., b_{d}}` be a basis for

`b _{i} b_{j} = sum_{k=1}^{d} a_{i,j,k} b_{k}.`

An associative algebra `A` is **nilpotent** if its **power series** terminates
at the trivial ideal of `A`; that is

` A > A ^{2} > ...> A^{c} > A^{c+1} = {0} `

where `A ^{j}` is the ideal of

For computational purposes we describe a nilpotent associative algebra by
a weighted basis and a description of the corresponding structure constants
table. A basis of a nilpotent associative algebra `A` is **weighted** if
there is a sequence of weights `(w _{1}, ..., w_{d})` so that

`A ^{j} = langleb_{i} midw_{i} geqj rangle.`

Note that `A A ^{j} = A^{j+1}` for every

Isomorphisms and Automorphisms

Let `A` be a finite dimensional nilpotent associative algebra over a
finite field. This package contains an implementation of the methods
in Eic07 which allow the determination of the automorphism group
`Aut(A)` and a **canonical form** `Can(A)`.

The automorphism group is given by generators and it represented as a
subgroup of `GL(dim(A), F)`. Also the order of `Aut(A)` is available.

A canonical form `Can(A)` for `A` is a nilpotent structure constants
table for `A` which is unique for the isomorphism type of `A`;
that is, two algebras `A` and `B` are isomorphic if and only if `Can(A)
= Can(B)` holds. Hence the canonical form can be used to solve the
isomorphism problem.

The modular isomorphism problem

The modular isomorphism problem asks whether `FG congFH` implies
that `G congH` for two `p`-groups `G` and `H` and `F` the field with `p`
elements. This problem is still open, despite various efforts towards
proving the claim or finding counterexamples to it.

Computational approaches have been used to investigate the modular isomorphism
problem. Based on an algorithm by Roggenkamp and Scott RS93, Wursthorn
Wur93 described an algorithm for checking the modular isomorphism
problem; that is, he described an algorithm for checking whether two modular
group algebras `FG` and `FH` are isomorphic. This algorithm has been
implemented in C by Wursthorn and has been used applied to the groups of
order dividing `2 ^{7}` without finding a counterexample, see BKRW99.

This package contains an implementation of the new algorithm described in
Eic07 for checking isomorphism of modular group algebras. It is based
on the fact that the Jacobson radical `J(FG)` is nilpotent if `FG` is a
modular group algebra. Hence the automorphism group and canonical form
algorithm of this package apply and can be used to solve the isomorphism
problem for modular group algebras.

The methods of this package have been used to check the modular isomorphism
problem for the groups of order dividing `3 ^{6}` and

A nilpotent quotient algorithm

Given a finitely presented associative algebra `A` over an arbitrary
field `F`, this package contains an algorithm to determine a nilpotent
structure constants table for the class-`c` nilpotent quotient of `A`.
See Eic11 for details on the underlying algorithm.

Kurosh Algebras

Let `F(d,F)` denote the free non-unital associative algebra on `d`
generators over the field `F`. Then

`A(d,n,F) = F(d,F) / langlelanglew ^{n} midw inF(d,F) ranglerangle`

is the **Kurosh Algebra** on `d` generators of exponent `n` over the field
`F`. Kurosh Algebras can be considered as an algebra-theoretic analogue to
Burnside groups.

This package contains a method that allows to determine `A(d,n,F)` for
given `d`, `n`, `F`. This can also be used to determine `A(d,n,F)` for all
fields of a given characteristic. We refer to Eic11 for details on
the algorithms.

This package also contains a database of Kurosh Algebras that have been determined with the methods of this package.

ModIsom manual

September 2018