Related papers: Finite soluble groups that act with fixity 2 or 3

A subgroup $H$ of a group $G$ is said to be an $IC\Phi$-subgroup of $G$ if $H \cap [H,G] \le \Phi(H)$. We analyze the structure of a finite group $G$ under the assumption that some given subgroups of $G$ are $IC\Phi$-subgroups of $G$. A new…

The principle result of this article is the determination of the possible finite subgroups of arithmetic lattices in U(2,1).

We study the class of groups having the property that every non-nilpotent subgroup is equal to its normalizer. These groups are either soluble or perfect. We completely describe the structure of soluble groups and finite perfect groups with…

We construct two finite groups of size $2^{365}\cdot 3^{105}\cdot 7^{104}$: a solvable group $G$ and a non-solvable group $H$, such that for every integer $n$ the groups have the same number of elements of order $n$. This answers a question…

We survey some results concerning finite group actions on products of spheres.

We prove that for every $\epsilon > 0$ there exists a $\delta > 0$ so that every group of order $n \geq 3$ has at least $\delta \log_{2} n/{(\log_{2} \log_{2} n)}^{3+\epsilon}$ conjugacy classes. This sharpens earlier results of Pyber and…

In this report we summarize this work, all finite simple groups $G$ can determined uniformly using their orders $|G|$ and the set $\pi_e(G)$ of their element orders.

In these notes we will survey recent results on various finitary approximation properties of infinite groups. We will discuss various restrictions on groups that are approximated for example by finite solvable groups or finite-dimensional…

We prove that the following problem is decidable: given a finite set of relations, decide whether this set admits a near-unanimity function.

We show that every finite group $G$ of size at least $3$ has a nilpotent subgroup of class at most $2$ and size at least $|G|^{1/32\log\log|G|}$. This answers a question of Pyber, and is essentially best possible.

For a group G we consider the set of natural numbers n for which the nth cohomology functor of G commutes with filtered colimit systems of coefficient modules. We find that for the large class of hierarchically decomposable groups there is…

A finitely generated solvable group with unbounded iterated identity is constructed.

A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory…

The genus spectrum of a finite group $G$ is the set of all $g\geq 2$ such that $G$ acts faithfully and orientation-preserving on a closed compact orientable surface of genus $g$. This article is an overview of some results relating the…

We show that a finite group which admits a faithful, smooth, orientation-preserving action on a homology 4-sphere, and in particular on the 4-sphere, is isomorphic to a subgroup of the orthogonal group SO(5), by explicitly determining the…

This is the third and last of three papers containing the complete proof that all finitely presented groups are QSF.

We determine the finite groups whose real irreducible representations have different degrees.

Two finite groups are said to have the same order type if for each positive integer $n$ both groups have the same number of elements of order $n$. In 1987 John G. Thompson asked if in this case the solvability of one group implies the…

This expository article revolves around the question to find short presentations of finite simple groups. This subject is one of the most active research areas of group theory in recent times. We bring together several known results on…

An algorithm for computing power conjugate presentations for finite soluble quotients of predetermined structure of finitely presented groups is described. Practical aspects of an implementation are discussed.