919 research outputs found
Faster Isomorphism for -Groups of Class 2 and Exponent
The group isomorphism problem determines whether two groups, given by their
Cayley tables, are isomorphic. For groups with order , an algorithm with
running time, attributed to Tarjan, was proposed in the
1970s [Mil78]. Despite the extensive study over the past decades, the current
best group isomorphism algorithm has an running time
[Ros13].
The isomorphism testing for -groups of (nilpotent) class 2 and exponent
has been identified as a major barrier to obtaining an time
algorithm for the group isomorphism problem. Although the -groups of class 2
and exponent have much simpler algebraic structures than general groups,
the best-known isomorphism testing algorithm for this group class also has an
running time.
In this paper, we present an isomorphism testing algorithm for -groups of
class 2 and exponent with running time for any
prime . Our result is based on a novel reduction to the skew-symmetric
matrix tuple isometry problem [IQ19]. To obtain the reduction, we develop
several tools for matrix space analysis, including a matrix space
individualization-refinement method and a characterization of the low rank
matrix spaces.Comment: Accepted to STOC 202
Testing isomorphism of graded algebras
We present a new algorithm to decide isomorphism between finite graded
algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it
runs in time polynomial in the order of the input algebras. We introduce
heuristics that often dramatically improve the performance of the algorithm and
report on an implementation in Magma
Coclass theory for nilpotent semigroups via their associated algebras
Coclass theory has been a highly successful approach towards the
investigation and classification of finite nilpotent groups. Here we suggest a
similar approach for finite nilpotent semigroups. This differs from the group
theory setting in that we additionally use certain algebras associated to the
considered semigroups. We propose a series of conjectures on our suggested
approach. If these become theorems, then this would reduce the classification
of nilpotent semigroups of a fixed coclass to a finite calculation. Our
conjectures are supported by the classification of nilpotent semigroups of
coclass 0 and 1. Computational experiments suggest that the conjectures also
hold for the nilpotent semigroups of coclass 2 and 3.Comment: 13 pages, 2 figure
A computer-based approach to the classification of nilpotent Lie algebras
We adopt the -group generation algorithm to classify small-dimensional
nilpotent Lie algebras over small fields. Using an implementation of this
algorithm, we list the nilpotent Lie algebras of dimension at most~9 over
\F_2 and those of dimension at most~7 over \F_3 and \F_5.Comment: submitte
The isomorphism problem for universal enveloping algebras of nilpotent Lie algebras
In this paper we study the isomorphism problem for the universal enveloping
algebras of nilpotent Lie algebras. We prove that if the characteristic of the
underlying field is not~2 or~3, then the isomorphism type of a nilpotent Lie
algebra of dimension at most~6 is determined by the isomorphism type of its
universal enveloping algebra. Examples show that the restriction on the
characteristic is necessary
Isomorphism in expanding families of indistinguishable groups
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. The recognized portions of their automorphism groups
are isomorphic, represented isomorphically on their abelianizations, and of
small index in their full automorphism groups. Nevertheless, there is a
polynomial-time algorithm to test for isomorphisms between these groups.Comment: 28 page
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