1,338 research outputs found
Fusion systems on bicyclic 2-groups
We classify all (saturated) fusion systems on bicyclic 2-groups. Here, a
bicyclic group is a product of two cyclic subgroups. This extends previous work
on fusion systems on metacyclic 2-groups (see [Craven-Glesser, 2012] and
[Sambale, 2012]). As an application we prove Olsson's Conjecture for all blocks
with bicyclic defect groups.Comment: 22 pages, shorted and some arguments replace
Wavelet analysis on symbolic sequences and two-fold de Bruijn sequences
The concept of symbolic sequences play important role in study of complex
systems. In the work we are interested in ultrametric structure of the set of
cyclic sequences naturally arising in theory of dynamical systems. Aimed at
construction of analytic and numerical methods for investigation of clusters we
introduce operator language on the space of symbolic sequences and propose an
approach based on wavelet analysis for study of the cluster hierarchy. The
analytic power of the approach is demonstrated by derivation of a formula for
counting of {\it two-fold de Bruijn sequences}, the extension of the notion of
de Bruijn sequences. Possible advantages of the developed description is also
discussed in context of applied
On the (non)existence of symplectic resolutions for imprimitive symplectic reflection groups
We study the existence of symplectic resolutions of quotient singularities
V/G where V is a symplectic vector space and G acts symplectically. Namely, we
classify the symplectically irreducible and imprimitive groups, excluding those
of the form where K < \SL_2(\C), for which the corresponding
quotient singularity admits a projective symplectic resolution. As a
consequence, for , we classify all quotient singularities
admitting a projective symplectic resolution which do not decompose as a
product of smaller-dimensional quotient singularities, except for at most four
explicit singularities, that occur in dimensions at most 10, for whom the
question of existence remains open.Comment: 21 page
Inverse zero-sum problems II
Let be an additive finite abelian group. A sequence over is called a
minimal zero-sum sequence if the sum of its terms is zero and no proper
subsequence has this property. Davenport's constant of is the maximum of
the lengths of the minimal zero-sum sequences over . Its value is well-known
for groups of rank two. We investigate the structure of minimal zero-sum
sequences of maximal length for groups of rank two. Assuming a well-supported
conjecture on this problem for groups of the form , we
determine the structure of these sequences for groups of rank two. Combining
our result and partial results on this conjecture, yields unconditional results
for certain groups of rank two.Comment: new version contains results related to Davenport's constant only;
other results will be described separatel
The modular isomorphism problem for finite -groups with a cyclic subgroup of index
Let be a prime number, be a finite -group and be a field of
characteristic . The Modular Isomorphism Problem (MIP) asks whether the
group algebra determines the group . Dealing with MIP, we investigated
a question whether the nilpotency class of a finite -group is determined by
its modular group algebra over the field of elements. We give a positive
answer to this question provided one of the following conditions holds: (i)
; (ii) \cl(G)=2; (iii) is cyclic; (iv) is a group of
maximal class and contains an abelian subgroup of index .Comment: 8 page
Fibered spherical 3-orbifolds
In early 1930s Seifert and Threlfall classified up to conjugacy the finite
subgroups of , this gives an algebraic classification of
orientable spherical 3-orbifolds. For the most part, spherical 3-orbifolds are
Seifert fibered. The underlying topological space and singular set of
non-fibered spherical 3-orbifolds were described by Dunbar. In this paper we
deal with the fibered case and in particular we give explicit formulae relating
the finite subgroups of with the invariants of the
corresponding fibered 3-orbifolds. This allows to deduce directly from the
algebraic classification topological properties of spherical 3-orbifolds.Comment: 27 pages, 6 figures. Several misprint corrected, improved expositio
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