Maximum orders of cyclic and abelian extendable actions on surfaces

Linear Extensions of Orders Invariant under Abelian Group Actions

Let G be an abelian group acting on a set X, and suppose that no element of G has any finite orbit of size greater than one. We show that every partial order on X invariant under G extends to a linear order on X also invariant under G. We then discuss extensions to linear preorders when the orbit condition is not met, and show that for any abelian group acting on a set X, there is a G-invariant...

Group Actions on Graphs, Maps and Surfaces with Maximum Symmetry

Finite $p$-groups and centralizers of non-cyclic abelian subgroups

A $p$-group $G$ is called a $mathcal{CAC}$-$p$-group if $C_G(H)/H$ is ‎cyclic for every non-cyclic abelian subgroup $H$ in $G$ with $Hnleq‎ ‎Z(G)$‎. ‎In this paper‎, ‎we give a complete classification of‎ ‎finite $mathcal{CAC}$-$p$-groups‎.

Rigidity of Trivial Actions of Abelian-by-cyclic Groups

The question of existence and stability of global fixed points for group actions has been studied in many different contexts. In the setting of actions of Lie groups it was shown by Lima [8] that n commuting vector fields on a genus g surface, Σg, of non-zero Euler characteristic have a common singularity. This implies that any action of the abelian Lie group R on Σg has a global fixed point. I...

Group Actions, Cyclic Coverings and Families of K3-surfaces

In this paper we describe six pencils of K3-surfaces which have large Picardnumber (15 ≤ ρ ≤ 20) and contain precisely five singular fibers: four have A-D-E singularities and one is non-reduced. In particular we describe these surfaces as cyclic coverings of the K3-surfaces of [BS]. In many cases using this description and latticetheory we are able to compute the exact Picard-number and to desc...