52 research outputs found
Polynomial-time Isomorphism Test for Groups with Abelian Sylow Towers
We consider the problem of testing isomorphism of groups of order n
given by Cayley tables. The trivial n^{log n} bound on the time
complexity for the general case has not been improved over the past
four decades. Recently, Babai et al. (following Babai et al. in SODA
2011) presented a polynomial-time algorithm for groups without abelian
normal subgroups, which suggests solvable groups as the hard case for
group isomorphism problem. Extending recent work by Le Gall (STACS
2009) and Qiao et al. (STACS 2011), in this paper we design a
polynomial-time algorithm to test isomorphism for the largest class of
solvable groups yet, namely groups with abelian Sylow towers, defined
as follows. A group G is said to possess a Sylow tower, if there
exists a normal series where each quotient is isomorphic to Sylow
subgroup of G. A group has an abelian Sylow tower if it has a Sylow
tower and all its Sylow subgroups are abelian. In fact, we are able
to compute the coset of isomorphisms of groups formed as coprime
extensions of an abelian group, by a group whose automorphism group is
known.
The mathematical tools required include representation theory,
Wedderburn\u27s theorem on semisimple algebras, and M.E. Harris\u27s 1980
work on p\u27-automorphisms of abelian p-groups. We use tools from the
theory of permutation group algorithms, and develop an algorithm for a
parameterized versin of the graph-isomorphism-hard setwise stabilizer
problem, which may be of independent interest
Algorithms for group isomorphism via group extensions and cohomology
The isomorphism problem for finite groups of order n (GpI) has long been
known to be solvable in time, but only recently were
polynomial-time algorithms designed for several interesting group classes.
Inspired by recent progress, we revisit the strategy for GpI via the extension
theory of groups.
The extension theory describes how a normal subgroup N is related to G/N via
G, and this naturally leads to a divide-and-conquer strategy that splits GpI
into two subproblems: one regarding group actions on other groups, and one
regarding group cohomology. When the normal subgroup N is abelian, this
strategy is well-known. Our first contribution is to extend this strategy to
handle the case when N is not necessarily abelian. This allows us to provide a
unified explanation of all recent polynomial-time algorithms for special group
classes.
Guided by this strategy, to make further progress on GpI, we consider
central-radical groups, proposed in Babai et al. (SODA 2011): the class of
groups such that G mod its center has no abelian normal subgroups. This class
is a natural extension of the group class considered by Babai et al. (ICALP
2012), namely those groups with no abelian normal subgroups. Following the
above strategy, we solve GpI in time for central-radical
groups, and in polynomial time for several prominent subclasses of
central-radical groups. We also solve GpI in time for
groups whose solvable normal subgroups are elementary abelian but not
necessarily central. As far as we are aware, this is the first time there have
been worst-case guarantees on a -time algorithm that tackles
both aspects of GpI---actions and cohomology---simultaneously.Comment: 54 pages + 14-page appendix. Significantly improved presentation,
with some new result
Beating the Generator-Enumeration Bound for -Group Isomorphism
We consider the group isomorphism problem: given two finite groups G and H
specified by their multiplication tables, decide if G cong H. For several
decades, the n^(log_p n + O(1)) generator-enumeration bound (where p is the
smallest prime dividing the order of the group) has been the best worst-case
result for general groups. In this work, we show the first improvement over the
generator-enumeration bound for p-groups, which are believed to be the hard
case of the group isomorphism problem. We start by giving a Turing reduction
from group isomorphism to n^((1 / 2) log_p n + O(1)) instances of p-group
composition-series isomorphism. By showing a Karp reduction from p-group
composition-series isomorphism to testing isomorphism of graphs of degree at
most p + O(1) and applying algorithms for testing isomorphism of graphs of
bounded degree, we obtain an n^(O(p)) time algorithm for p-group
composition-series isomorphism. Combining these two results yields an algorithm
for p-group isomorphism that takes at most n^((1 / 2) log_p n + O(p)) time.
This algorithm is faster than generator-enumeration when p is small and slower
when p is large. Choosing the faster algorithm based on p and n yields an upper
bound of n^((1 / 2 + o(1)) log n) for p-group isomorphism.Comment: 15 pages. This is an updated and improved version of the results for
p-groups in arXiv:1205.0642 and TR11-052 in ECC
Polynomial-Time Isomorphism Test of Groups that are Tame Extensions
We give new polynomial-time algorithms for testing isomorphism of a class of
groups given by multiplication tables (GpI). Two results (Cannon & Holt, J.
Symb. Comput. 2003; Babai, Codenotti & Qiao, ICALP 2012) imply that GpI reduces
to the following: given groups G, H with characteristic subgroups of the same
type and isomorphic to , and given the coset of isomorphisms
, compute Iso(G, H) in time poly(|G|).
Babai & Qiao (STACS 2012) solved this problem when a Sylow p-subgroup of
is trivial. In this paper, we solve the preceding problem in
the so-called "tame" case, i.e., when a Sylow p-subgroup of
is cyclic, dihedral, semi-dihedral, or generalized quaternion. These cases
correspond exactly to the group algebra
being of tame type, as in the
celebrated tame-wild dichotomy in representation theory. We then solve new
cases of GpI in polynomial time.
Our result relies crucially on the divide-and-conquer strategy proposed
earlier by the authors (CCC 2014), which splits GpI into two problems, one on
group actions (representations), and one on group cohomology. Based on this
strategy, we combine permutation group and representation algorithms with new
mathematical results, including bounds on the number of indecomposable
representations of groups in the tame case, and on the size of their cohomology
groups.
Finally, we note that when a group extension is not tame, the preceding
bounds do not hold. This suggests a precise sense in which the tame-wild
dichotomy from representation theory may also be a dividing line between the
(currently) easy and hard instances of GpI.Comment: 23 page
Parallel algorithms for solvable permutation groups
AbstractA number of basic problems involving solvable and nilpotent permutation groups are shown to have fast parallel solutions. Testing solvability is in NC as well as, for solvable groups, finding order, testing membership, finding centralizers, finding centers, finding the derived series and finding a composition series. Additionally, for nilpotent groups, one can, in NC, find a central composition series, and find pointwise stabilizers of sets. The latter is applied to an instance of graph isomorphism. A useful tool is the observation that the problem of finding the smallest subspace containing a given set of vectors and closed under a given set of linear transformations (all over a small field) belongs to NC
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