18,576 research outputs found
On the existence of zero-sum subsequences of distinct lengths
In this paper, we obtain a characterization of short normal sequences over a
finite Abelian p-group, thus answering positively a conjecture of Gao for a
variety of such groups. Our main result is deduced from a theorem of Alon,
Friedland and Kalai, originally proved so as to study the existence of regular
subgraphs in almost regular graphs. In the special case of elementary p-groups,
Gao's conjecture is solved using Alon's Combinatorial Nullstellensatz. To
conclude, we show that, assuming every integer satisfies Property B, this
conjecture holds in the case of finite Abelian groups of rank two.Comment: 10 pages, to appear in Rocky Mountain Journal of Mathematic
The Dynamics of Group Codes: Dual Abelian Group Codes and Systems
Fundamental results concerning the dynamics of abelian group codes
(behaviors) and their duals are developed. Duals of sequence spaces over
locally compact abelian groups may be defined via Pontryagin duality; dual
group codes are orthogonal subgroups of dual sequence spaces. The dual of a
complete code or system is finite, and the dual of a Laurent code or system is
(anti-)Laurent. If C and C^\perp are dual codes, then the state spaces of C act
as the character groups of the state spaces of C^\perp. The controllability
properties of C are the observability properties of C^\perp. In particular, C
is (strongly) controllable if and only if C^\perp is (strongly) observable, and
the controller memory of C is the observer memory of C^\perp. The controller
granules of C act as the character groups of the observer granules of C^\perp.
Examples of minimal observer-form encoder and syndrome-former constructions are
given. Finally, every observer granule of C is an "end-around" controller
granule of C.Comment: 30 pages, 11 figures. To appear in IEEE Trans. Inform. Theory, 200
Convergent sequences in discrete groups
We prove that a finitely generated group contains a sequence of non-trivial
elements which converge to the identity in every compact homomorphic image if
and only if the group is not virtually abelian.Comment: 10 pages, no figures; in v2 some material on Jordan's Theorem and Chu
duality added; v3 updated version, to appear in the Canadian Math. Bulleti
Exact sequences of tensor categories
We introduce the notions of normal tensor functor and exact sequence of
tensor categories. We show that exact sequences of tensor categories generalize
strictly exact sequences of Hopf algebras as defined by Schneider, and in
particular, exact sequences of (finite) groups. We classify exact sequences of
tensor categories C' -> C -> C'' (such that C' is finite) in terms of normal
faithful Hopf monads on C'' and also, in terms of self-trivializing commutative
algebras in the center of C. More generally, we show that, given any dominant
tensor functor C -> D admitting an exact (right or left) adjoint there exists a
canonical commutative algebra A in the center of C such that F is tensor
equivalent to the free module functor C -> mod_C A, where mod_C A denotes the
category of A-modules in C endowed with a monoidal structure defined using the
half-braiding of A. We re-interpret equivariantization under a finite group
action on a tensor category and, in particular, the modularization
construction, in terms of exact sequences, Hopf monads and commutative central
algebras. As an application, we prove that a braided fusion category whose
dimension is odd and square-free is equivalent, as a fusion category, to the
representation category of a group.Comment: 39 page
Reduction of UNil for finite groups with normal abelian Sylow 2-subgroup
Let F be a finite group with a Sylow 2-subgroup S that is normal and abelian.
Using hyperelementary induction and cartesian squares, we prove that Cappell's
unitary nilpotent groups UNil_*(Z[F];Z[F],Z[F]) have an induced isomorphism to
the quotient of UNil_*(Z[S];Z[S],Z[S]) by the action of the group F/S. In
particular, any finite group F of odd order has the same UNil-groups as the
trivial group. The broader scope is the study of the L-theory of virtually
cyclic groups, based on the Farrell--Jones isomorphism conjecture. We obtain
partial information on these UNil when S is a finite abelian 2-group and when S
is a special 2-group.Comment: 29 pages, revision of decorations, correction of Homological
Reductio
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