Automorphisms of Cayley graphs on generalised dicyclic groups

A graph is called a GRR if its automorphism group acts regularly on its vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that there are only two infinite families of finite groups that do not admit GRRs : abelian groups and generalised dicyclic groups. Indeed, any Cayley graph on such a group admits specific additional graph automorphisms that depend only on the group. Recently, Dobson and the last two authors showed that almost all Cayley graphs on abelian groups admit no automorphisms other than these obvious necessary ones. In this paper, we prove the analogous result for Cayley graphs on the remaining family of exceptional groups: generalised dicyclic groups.Comment: 18 page

On finite pp-groups whose automorphisms are all central

An automorphism $\alpha$ of a group $G$ is said to be central if $\alpha$ commutes with every inner automorphism of $G$. We construct a family of non-special finite $p$-groups having abelian automorphism groups. These groups provide counter examples to a conjecture of A. Mahalanobis [Israel J. Math., {\bf 165} (2008), 161 - 187]. We also construct a family of finite $p$-groups having non-abelian automorphism groups and all automorphisms central. This solves a problem of I. Malinowska [Advances in group theory, Aracne Editrice, Rome 2002, 111-127].Comment: 11 pages, Counter examples to a conjecture from [Israel J. Math., {\bf 165} (2008), 161 - 187]; This paper will appear in Israel J. Math. in 201

On the Galoisian Structure of Heisenberg Indeterminacy Principle

We revisit Heisenberg indeterminacy principle in the light of the Galois-Grothendieck theory for the case of finite abelian Galois extensions. In this restricted framework, the Galois-Grothendieck duality between finite K-algebras split by a Galois extension L and finite Gal(L:K)-sets can be reformulated as a Pontryagin-like duality between two abelian groups. We then define a Galoisian quantum theory in which the Heisenberg indeterminacy principle between conjugate canonical variables can be understood as a form of Galoisian duality: the larger the group of automorphisms H (a subgroup of G) of the states in a G-set O = G/H, the smaller the ``conjugate'' observable algebra that can be consistently valuated on such states. We then argue that this Galois indeterminacy principle can be understood as a particular case of the Heisenberg indeterminacy principle formulated in terms of the notion of entropic indeterminacy. Finally, we argue that states endowed with a group of automorphisms H can be interpreted as squeezed coherent states, i.e. as states that minimize the Heisenberg indeterminacy relations

On the Brauer groups of symmetries of abelian Dijkgraaf-Witten theories

Symmetries of three-dimensional topological field theories are naturally defined in terms of invertible topological surface defects. Symmetry groups are thus Brauer-Picard groups. We present a gauge theoretic realization of all symmetries of abelian Dijkgraaf-Witten theories. The symmetry group for a Dijkgraaf-Witten theory with gauge group a finite abelian group $A$, and with vanishing 3-cocycle, is generated by group automorphisms of $A$, by automorphisms of the trivial Chern-Simons 2-gerbe on the stack of $A$-bundles, and by partial e-m dualities. We show that transmission functors naturally extracted from extended topological field theories with surface defects give a physical realization of the bijection between invertible bimodule categories of a fusion category and braided auto-equivalences of its Drinfeld center. The latter provides the labels for bulk Wilson lines; it follows that a symmetry is completely characterized by its action on bulk Wilson lines.Comment: 21 pages, 9 figures. v2: Minor changes, typos corrected and references added. v3: Typos correcte