Related papers: Fixed point conditions for non-coprime actions

By an additive action on an algebraic variety $X$ over $\mathbb{C}$, we mean an action of the group $\mathbb{G}_a^n = \mathbb{C}^n $ on $X$ with an open orbit. We study limit points of one-dimensional subgroups of $\mathbb{G}_a^n$ for…

Given an action by a finite quantum group $\mathbb{G}$ on a von Neumann algebra $M$, we prove that a number of familiar $W^*$ properties are equivalent for $M$ and the fixed-point algebra $M^{\mathbb{G}}$ (i.e. hold or not simultaneously…

A group $G$ is said to be totally $k$-closed for a positive integer $k$ if, in each of its faithful permutation representations on a set $\Omega^k$, $G$ is the largest subgroup of the symmetric group $\operatorname{Sym}(\Omega)$ that…

In this paper we consider the finite groups that act fiber- and orientation-preservingly on closed, compact, and orientable Seifert manifolds that fiber over an orientable base space. We establish a method of constructing such group actions…

In this paper, we investigate the existence of fixed-point-free automorphisms for finite-dimensional Lie algebras. By a result of Jacobson, a Lie algebra admitting a fixed-point-free automorphism is solvable. We prove that such a Lie…

For a p-group G admitting an automorphism $\phi$ of order $p^n$ with exactly $p^m$ fixed points such that $\phi^{p^{n-1}}$ has exactly $p^k$ fixed points, we prove that G has a fully-invariant subgroup of m-bounded nilpotency class with…

We provide the first known example of a finite group action on an oriented surface $T$ that is free, orientation-preserving, and does not extend to an arbitrary (in particular, possibly non-free) orientation-preserving action on any compact…

For a finite group $G$ not of prime power order, Oliver (1996) has answered the question which manifolds occur as the fixed point sets of smooth actions of $G$ on disks (resp., Euclidean spaces). We extend Oliver's result to compact (resp.,…

Let $N$ be a nilpotent group normal in a group $G$. Suppose that $G$ acts transitively upon the points of a finite non-Desarguesian projective plane $\mathcal{P}$. We prove that, if $\mathcal{P}$ has square order, then $N$ must act…

In this paper, we study the action on C^n of any group G of holomorphic diffeomorphisms (automorphisms) of C^n fixing 0. Suppose that there is x in C^n, having an orbit which generates C^n and also E(x)=C^n, where E(x) is the vector space…

In this second part we prove that, if $G$ is one of the groups $\mathrm{PSL}_2(q)$ with $q>5$ and $q\equiv 5\pmod {24}$ or $q\equiv 13 \pmod{24}$, then the fundamental group of every acyclic $2$-dimensional, fixed point free and finite…

Let G be a finitely generated group having the property that any action of any finite-index subgroup of G by homeomorphisms of the circle must have a finite orbit. (By a theorem of E.Ghys, lattices in simple Lie groups of real rank at least…

In this paper, we are concerned with the direct product $G=\pi_1(\Sigma)\times \Z^k$ for $\Sigma$ a compact orientable surface with negative Euler characteristic, and give a complete classification of its fixed subgroups of automorphisms.…

Let $G$ be a permutation group acting on a finite set $\Omega$ of cardinality $n$. The number of orbits of the induced action of $G$ on the set $\Omega_m$ of all size $m$ subsets of $\Omega$ satisfies the trivial inequalities…

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the…

Let $K$ be an algebraically closed field. Let $G$ be a non-trivial connected unipotent group, which acts effectively on an affine variety $X.$ Then every non-empty component $R$ of the set of fixed points of $G$ is a $K$-uniruled variety,…

This is a follow-up to a paper with the same title and by the same authors. In that paper, all groups were assumed to be abelian, and we are now aiming to generalize the results to nonabelian groups. The motivating point is Pedersen's…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. We provide a complete classification of a finite group $G$ in which every maximal $A$-invariant subgroup containing the normalizer of some $A$-invariant…

We consider Abelian-by-cyclic groups for which the cyclic factor acts by hyperbolic automorphisms on the Abelian subgroup. We show that if such a group acts faithfully by $C^1$ diffeomorphisms of the closed interval with no global fixed…

Let $p>3$ be a prime. We show that if $G$ is a finite group with $p$-rank equal to 2, then $G$ involves $Qd(p)$ if and only if $G$ $p'$-involves $Qd(p)$. This allows us to use a version of Glauberman's ZJ-theorem to give a more direct…