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<h1>1 Introduction</h1>

This package contains various algorithms related to finite dimensional
nilpotent associative algebras. It also contains many group-theoretical
functions related to the Modular Isomorphism Problem.
We first give a brief introduction to finite dimensional
nilpotent algebras and then an overview of the main algorithms.


<hr>Associative algebras and nilpotency

Let A be an associative algebra of dimension d over a field F.
Let {b1, …, bd} be a basis for A. We identify the
element x1 b1 + …+ xd bd of A with the element
(x1, …, xd) of Fd. The multiplication of A can then
be described by a structure constants table: a 3-dimensional array
with entries ai,j,k ∈ F satisfying that
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center">bi bj = </td><td nowrap="nowrap" align="center">d ∑ k=1 </td><td nowrap="nowrap" align="center">ai,j,k bk·</td></tr></table></td></tr></table>


An associative algebra A is nilpotent if its power series terminates
at the trivial ideal of A; that is
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> A > A2 > … > Ac > Ac+1 = {0} </td></tr></table></td></tr></table>
where Aj is the ideal of A generated by all products of length
at least j. The length c of the power series is also called the
class of A and the dimension of A/A2 is the rank of A. Note
that A is generated by dim(A/A2) elements. Clearly, A does not
contain a multiplicative identity.


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 (w1, …, wd) so that
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center">Aj = 〈bi | wi ≥ j 〉·</td></tr></table></td></tr></table>
Note that A Aj = Aj+1 for every j. Thus it is possible to choose
all basis elements of weight at least 2 so that bi = bk bl holds for
some k and l, where bk is of weight 1 and bl is of weight wi−1.
This feature allows an effective description of A via a nilpotent
structure constants table. This contains the structure constants
ai,j,k for all i with wi = 1 and 1 ≤ j,k ≤ d. For i
with wi > 1 it either contains a description as bi = bk bl or the
structure constants ai,j,k for 1 ≤ j,k ≤ d. It may also
contain both or some partial overlap of these informations.


<hr>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 <a href="biblio.htm#Eic07"><cite>Eic07</cite></a> which allow the determination of the automorphism group
Aut(A) and a canonical form Can(A).

The automorphism group is given by generators and is 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.


<hr>The Modular Isomorphism Problem

The modular isomorphism problem asks whether an isomorphism of algebras Fp G ≅ Fp H implies
an isomorphism of groups G ≅ H for two p-groups G and H and Fp the field with p
elements. This problem was open for a long time until first counterexamples
for the prime p=2 were found in <a href="biblio.htm#GLMdR22"><cite>GLMdR22</cite></a>. It remains open for odd
primes and many other interesting classes of groups.

Computational approaches have been used to investigate the modular isomorphism
problem. Based on an algorithm by Roggenkamp and Scott <a href="biblio.htm#RS93"><cite>RS93</cite></a>, Wursthorn
<a href="biblio.htm#Wur93"><cite>Wur93</cite></a> described an algorithm for checking the modular isomorphism
problem; that is, he described an algorithm for checking whether two modular
group algebras Fp G and Fp H are isomorphic, where G and H are finite
p-groups. This algorithm has been
implemented in C by Wursthorn and has been applied to the groups of
order dividing 27 without finding a counterexample, see <a href="biblio.htm#BKRW99"><cite>BKRW99</cite></a>.
The implementation of Wursthorn appears lost, but is in any case not publicly
available.


This package contains an implementation of the new algorithm described in
<a href="biblio.htm#Eic07"><cite>Eic07</cite></a> 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 for G a finite p-group and FG is isomorphic to FH if and only if the radicals
J(FG) and J(FH) are isomorphic. 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 of finite p-groups. Note that in this setting the Jacobson radical of the group algebra FG equals its augmentation ideal.

The methods of this package have been used to study the modular isomorphism
problem for the groups of order dividing 36 and 28 (<a href="biblio.htm#Eic07"><cite>Eic07</cite></a>) and
for the groups of order 29 (<a href="biblio.htm#EKo11"><cite>EKo11</cite></a>). It was later used to study also
groups of order 37 and 56 (<a href="biblio.htm#MM22"><cite>MM22</cite></a>).


A property of a group G is called F-invariant, if an isomorphism of
F-algebras FG ≅ FH implies the same property for H. In the context
of the Modular Isomorphism Problem, if G is a finite p-group, then an
Fp-invariant is simply called
invariant. Many invariants of G are known and the package provides
functions for them, as well as programs which easily allow to compare all
the implemented invariants quickly for a given list of groups.


It also remains open, if replacing the field Fp in the Modular Isomorphism
Problem with a bigger field of characteristic p will change the outcome
of the problem for a given pair of groups. The package includes several
functions which allow to investigate this question by applying the algorithm
for the same groups varying the field.


<hr>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,
i.e. the algebra A/Ac+1.
See <a href="biblio.htm#Eic11"><cite>Eic11</cite></a> for details on the underlying algorithm.


<hr>Kurosh Algebras

Let F(d,F) denote the free non-unital associative algebra on d
generators over the field F. Then
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center">A(d,n,F) = F(d,F) / 〈〈wn | w ∈ F(d,F) 〉〉</td></tr></table></td></tr></table>
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 <a href="biblio.htm#Eic11"><cite>Eic11</cite></a> for details on
the algorithms.

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


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