20 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
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
Efficient Isomorphism Testing for a Class of Group Extensions
The group isomorphism problem asks whether two given groups are isomorphic or
not. Whereas the case where both groups are abelian is well understood and can
be solved efficiently, very little is known about the complexity of isomorphism
testing for nonabelian groups. In this paper we study this problem for a class
of groups corresponding to one of the simplest ways of constructing nonabelian
groups from abelian groups: the groups that are extensions of an abelian group
A by a cyclic group of order m. We present an efficient algorithm solving the
group isomorphism problem for all the groups of this class such that the order
of A is coprime with m. More precisely, our algorithm runs in time almost
linear in the orders of the input groups and works in the general setting where
the groups are given as black-boxes.Comment: 17 pages, accepted to the STACS 2009 conferenc
On the isomorphism problem for certain -groups
We consider 9 infinite families of finite -groups, for a prime, and we
settle the isomorphism problem that arises when the parameters that define
these groups are modified