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Quelle intro.tex
Sprache: Latech
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\Chapter{Introduction}
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.
\medskip
{\bf Associative algebras and nilpotency}
Let $A$ be an associative algebra of dimension $d$ over a field $F$.
Let $\{b_1, \ldots, b_d\}$ be a basis for $A$. We identify the
element $x_1 b_1 + \ldots + x_d b_d$ of $A$ with the element
$(x_1, \ldots, x_d)$ of $F^d$. The multiplication of $A$ can then
be described by a *structure constants table*: a 3-dimensional array
with entries $a_{i,j,k} \in F$ satisfying that
$$b_i b_j = \sum_{k=1}^d a_{i,j,k} b_k.$$
\medskip
An associative algebra $A$ is *nilpotent* if its *power series* terminates
at the trivial ideal of $A$; that is
$$
A > A^2 > \ldots > A^c > A^{c+1} = \{0\} $$
where $A^j$ 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 $A/A^2$ is the *rank* of $A$. Note
that $A$ is generated by $dim(A/A^2)$ elements. Clearly, $A$ does not
contain a multiplicative identity.
\medskip
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 $(w_1, \ldots, w_d)$ so that
$$A^j = \langle b_i \mid w_i \geq j \rangle.$$
Note that $A A^j = A^{j+1}$ for every $j$. Thus it is possible to choose
all basis elements of weight at least 2 so that $b_i = b_k b_l$ holds for
some $k$ and $l$, where $b_k$ is of weight 1 and $b_l$ is of weight $w_i-1$.
This feature allows an effective description of $A$ via a *nilpotent
structure constants table*. This contains the structure constants
$a_{i,j,k}$ for all $i$ with $w_i = 1$ and $1 \leq j,k \leq d$. For $i$
with $w_i > 1$ it either contains a description as $b_i = b_k b_l$ or the
structure constants $a_{i,j,k}$ for $1 \leq j,k \leq d$. It may also
contain both or some partial overlap of these informations.
\medskip
{\bf 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 \cite{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 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.
\medskip
{\bf The Modular Isomorphism Problem}
The modular isomorphism problem asks whether an isomorphism of algebras $\F_p G \cong \F_p H$ implies
an isomorphism of groups $G \cong H$ for two $p$-groups $G$ and $H$ and $\F_p$ the field with $p$
elements. This problem was open for a long time until first counterexamples
for the prime $p=2$ were found in \cite{GLMdR22}. 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 \cite{RS93}, Wursthorn
\cite{Wur93} described an algorithm for checking the modular isomorphism
problem; that is, he described an algorithm for checking whether two modular
group algebras $\F_p G$ and $\F_p H$ 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 $2^7$ without finding a counterexample, see \cite{BKRW99}.
The implementation of Wursthorn appears lost, but is in any case not publicly
available.
\medskip
This package contains an implementation of the new algorithm described in
\cite{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 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 $3^6$ and $2^8$ (\cite{Eic07}) and
for the groups of order $2^9$ (\cite{EKo11}). It was later used to study also
groups of order $3^7$ and $5^6$ (\cite{MM22}).
\medskip
A property of a group $G$ is called *$F$-invariant*, if an isomorphism of
$F$-algebras $FG \cong FH$ implies the same property for $H$. In the context
of the Modular Isomorphism Problem, if $G$ is a finite $p$-group, then an
$\F_p$-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.
\medskip
It also remains open, if replacing the field $\F_p$ 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.
\medskip
{\bf 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 $A/A^{c+1}$.
See \cite{Eic11} for details on the underlying algorithm.
\medskip
{\bf Kurosh Algebras}
Let $F(d,F)$ denote the free non-unital associative algebra on $d$
generators over the field $F$. Then
$$A(d,n,F) = F(d,F) / \langle \langle w^n \mid w \in F(d,F) \rangle \rangle$$
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 \cite{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.
¤ Dauer der Verarbeitung: 0.1 Sekunden
(vorverarbeitet)
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*© Formatika GbR, Deutschland
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2026-03-28
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