scholarly journals AUTOMORPHISMS OF PARTIALLY COMMUTATIVE GROUPS II: COMBINATORIAL SUBGROUPS

2012 ◽  
Vol 22 (07) ◽  
pp. 1250074 ◽  
Author(s):  
ANDREW J. DUNCAN ◽  
VLADIMIR N. REMESLENNIKOV
Keyword(s):  
Automorphism Group ◽  
Decomposition Theorem ◽  
Direct Decomposition ◽  
Connected Components ◽  
Artin Group ◽  
Special Case

We define several "standard" subgroups of the automorphism group Aut (G) of a partially commutative (right-angled Artin) group and use these standard subgroups to describe decompositions of Aut (G). If C is the commutation graph of G, we show how Aut (G) decomposes in terms of the connected components of C: obtaining a particularly clear decomposition theorem in the special case where C has no isolated vertices. If C has no vertices of a type we call dominated then we give a semi-direct decomposition of Aut (G) into a subgroup of locally conjugating automorphisms by the subgroup stabilizing a certain lattice of "admissible subsets" of the vertices of C. We then characterize those graphs for which Aut (G) is a product (not necessarily semi-direct) of two such subgroups.

scholarly journals TOWARD A CLASSIFICATION OF FINITE QUANDLES

2008 ◽  
Vol 17 (04) ◽  
pp. 511-520 ◽  
Author(s):  
G. EHRMAN ◽  
A. GURPINAR ◽  
M. THIBAULT ◽  
D. N. YETTER
Keyword(s):  
Automorphism Group ◽  
Structure Theorem ◽  
Decomposition Theorem ◽  
New Approach ◽  
Inner Automorphism ◽  
Fourth Author ◽  
Student Team

This paper summarizes substantive new results derived by a student team (the first three authors) under the direction of the fourth author at the 2005 session of the KSU REU "Brainstorming and Barnstorming". The main results are a decomposition theorem for quandles in terms of an operation of "semidisjoint union" showing that all finite quandles canonically decompose via iterated semidisjoint unions into connected subquandles, and a structure theorem for finite connected quandles with prescribed inner automorphism group. The latter theorem suggests a new approach to the classification of finite connected quandles.

scholarly journals An Edmonds-Gallai-Type Decomposition for the j-Restricted k-Matching Problem

10.37236/7837 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Yanjun Li ◽  
Jácint Szabó
Keyword(s):  
Positive Integer ◽  
Undirected Graph ◽  
Decomposition Theorem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Alternative Proof ◽  
Connected Components ◽  
Matching Problem ◽  
Np Hardness ◽  
Type Decomposition

Given a non-negative integer $j$ and a positive integer $k$, a $j$-restricted $k$-matching in a simple undirected graph is a $k$-matching, so that each of its connected components has at least $j+1$ edges. The maximum non-negative node weighted $j$-restricted $k$-matching problem was recently studied by Li who gave a polynomial-time algorithm and a min-max theorem for $0 \leqslant j < k$, and also proved the NP-hardness of the problem with unit node weights and $2 \leqslant k \leqslant j$. In this paper we derive an Edmonds–Gallai-type decomposition theorem for the $j$-restricted $k$-matching problem with $0 \leqslant j < k$, using the analogous decomposition for $k$-piece packings given by Janata, Loebl and Szabó, and give an alternative proof to the min-max theorem of Li.

scholarly journals Sets in homotopy type theory

2015 ◽  
Vol 25 (5) ◽  
pp. 1172-1202 ◽  
Author(s):  
EGBERT RIJKE ◽  
BAS SPITTERS
Keyword(s):  
Homotopy Type ◽  
Type Theory ◽  
Connected Components ◽  
Homotopy Type Theory ◽  
Universe Of Sets ◽  
Univalent Foundations ◽  
Subobject Classifier ◽  
Theoretical Foundations ◽  
Special Case

Homotopy type theory may be seen as an internal language for the ∞-category of weak ∞-groupoids. Moreover, weak ∞-groupoids model the univalence axiom. Voevodsky proposes this (language for) weak ∞-groupoids as a new foundation for Mathematics called the univalent foundations. It includes the sets as weak ∞-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those ‘discrete’ groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of ‘elementary’ of ∞-toposes. We prove that sets in homotopy type theory form a ΠW-pretopos. This is similar to the fact that the 0-truncation of an ∞-topos is a topos. We show that both a subobject classifier and a 0-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover the 0-object classifier for sets is a function between 1-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets.

Generalised Bohr compactification and model-theoretic connected components

2016 ◽  
Vol 163 (2) ◽  
pp. 219-249 ◽  
Author(s):  
KRZYSZTOF KRUPIŃSKI ◽  
ANAND PILLAY
Keyword(s):  
Normal Subgroup ◽  
Discrete Group ◽  
Topological Dynamics ◽  
Discrete Case ◽  
Connected Components ◽  
Bohr Compactification ◽  
First Order ◽  
Hausdorff Group ◽  
The Given ◽  
Special Case

AbstractFor a group G first order definable in a structure M, we continue the study of the “definable topological dynamics” of G (from [9] for example). The special case when all subsets of G are definable in the given structure M is simply the usual topological dynamics of the discrete group G; in particular, in this case, the words “externally definable” and “definable” can be removed in the results described below.Here we consider the mutual interactions of three notions or objects: a certain model-theoretic invariant G*/(G*)000M of G, which appears to be “new” in the classical discrete case and of which we give a direct description in the paper; the [externally definable] generalised Bohr compactification of G; [externally definable] strong amenability. Among other things, we essentially prove: (i) the “new” invariant G*/(G*)000M lies in between the externally definable generalised Bohr compactification and the definable Bohr compactification, and these all coincide when G is definably strongly amenable and all types in SG(M) are definable; (ii) the kernel of the surjective homomorphism from G*/(G*)000M to the definable Bohr compactification has naturally the structure of the quotient of a compact (Hausdorff) group by a dense normal subgroup; (iii) when Th(M) is NIP, then G is [externally] definably amenable iff it is externally definably strongly amenable.In the situation when all types in SG(M) are definable, one can just work with the definable (instead of externally definable) objects in the above results.

scholarly journals Labelled seeds and the mutation group

2016 ◽  
Vol 163 (2) ◽  
pp. 193-217
Author(s):  
ALASTAIR KING ◽  
MATTHEW PRESSLAND
Keyword(s):  
Automorphism Group ◽  
Homogeneous Space ◽  
Equivalence Relation ◽  
Cluster Algebra ◽  
Equivalence Relations ◽  
Connected Components ◽  
Model Field ◽  
Regular Equivalence ◽  
Mutation Group

AbstractWe study the set${\mathcal{S}}$of labelled seeds of a cluster algebra of rankninside a field${\mathcal{F}}$as a homogeneous space for the groupMnof (globally defined) mutations and relabellings. Regular equivalence relations on${\mathcal{S}}$are associated to subgroupsWof AutMn(${\mathcal{S}}$), and we thus obtain groupoidsW\${\mathcal{S}}$. We show that for two natural choices of equivalence relation, the corresponding groupsWcandW+act on${\mathcal{F}}$, and the groupoidsWc\${\mathcal{S}}$andW+\${\mathcal{S}}$act on the model field${\mathcal{K}}$=ℚ(x1,. . .,xn). The groupoidW+\${\mathcal{S}}$is equivalent to Fock–Goncharov's cluster modular groupoid. Moreover,Wcis isomorphic to the group of cluster automorphisms, andW+to the subgroup of direct cluster automorphisms, in the sense of Assem–Schiffler–Shramchenko.We also prove that, for mutation classes whose seeds have mutation finite quivers, the stabiliser of a labelled seed underMndetermines the quiver of the seed up to ‘similarity’, meaning up to taking opposites of some of the connected components. Consequently, the subgroupWcis the entire automorphism group of${\mathcal{S}}$in these cases.

On the Automorphism Group of a Finite p-Group with a Small Central Quotient

1980 ◽  
Vol 32 (5) ◽  
pp. 1168-1176 ◽  
Author(s):  
Richard M. Davitt
Keyword(s):  
Automorphism Group ◽  
Central Quotient ◽  
Special Case

In recent years there has been considerable interest in the conjecture that |G| divides |Aut G| for all finite non-cyclic p-groups G of order greater than p2. In particular, the conjecture has been established for a considerable number of (not necessarily distinct) classes of finite p-groups ([6], [7], [8], [9], [15], [16]); additionally, results have been obtained, often using homological methods, which permit reductions in any attempt to establish the overall conjecture ([5], [10], [13], [15]). In the former case, the p-groups G have generally been regular p-groups (see, for example, [6]) and the prime p = 2 has either been excluded (see, for example, [8]) or treated as a special case (as in [9]).It is the purpose of this paper to establish the conjecture for the class of all p-groups G where |G: Z(G)| ≦ p4 with no restrictions on the prime p.

scholarly journals Cluster complexes via semi-invariants

2009 ◽  
Vol 145 (4) ◽  
pp. 1001-1034 ◽  
Author(s):  
Kiyoshi Igusa ◽  
Kent Orr ◽  
Gordana Todorov ◽  
Jerzy Weyman
Keyword(s):  
Fundamental Theorem ◽  
Decomposition Theorem ◽  
Canonical Decomposition ◽  
Cluster Complexes ◽  
Dynkin Quivers ◽  
Saturation Theorem ◽  
Representation Spaces ◽  
Special Case

AbstractWe define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the first fundamental theorem, the saturation theorem and the canonical decomposition theorem. In the special case of Dynkin quivers with n vertices, this gives the fundamental interrelationship between supports of the semi-invariants and the tilting triangulation of the (n−1)-sphere.

scholarly journals A Note on Automorphisms of the Infinite-Dimensional Hypercube Graph

10.37236/2749 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Mark Pankov
Keyword(s):  
Automorphism Group ◽  
Connected Components ◽  
Infinite Dimensional ◽  
The Family ◽  
Vertex Set

We define the infinite-dimensional hypercube graph $H_{{\aleph}_{0}}$ as a graph whose vertex set is formed by the so-called singular subsets of ${\mathbb Z}\setminus\{0\}$. This graph is not connected, but it has isomorphic connected components. We show that the restrictions of its automorphisms to the connected components  are induced by permutations on ${\mathbb Z}\setminus\{0\}$ preserving the family of singular subsets. As an application, we describe the automorphism group of the connected components.

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