Related papers: Finite p-groups with a Frobenius group of automorp…

We associate a graph $\Gamma_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G | \left<x,y\right> \text{is cyclic for all} y\in G\}$, and…

For $\beta\in{\mathbb Z}$, let $G(\beta)=\langle A,B\,|\, A^{[A,B]}=A,\, B^{[B,A]}=B^\beta\rangle$ be the infinite Macdonald group, and set $C=[A,B]$. Then $G(\beta)$ is a nilpotent polycyclic group of the form $\langle…

We show that $\omega$-categorical rings with NIP are nilpotent-by-finite. We prove that an $\omega$-categorical group with NIP and fsg is nilpotent-by-finite. We also notice that an $\omega$-categorical group with at least one strongly…

Let $p$ be a fixed prime number and let $R$ denote a uniserial $p$-adic space group of dimension $d_x=(p-1)p^{x-1}$ and with cyclic point group of order $p^x$. In this short note we prove that all the quotients of $R$ of size bigger than or…

For each prime p, we exhibit pairs of p-groups all of whose integral cohomology groups are isomorphic. The method used involves very little calculation. The groups are exhibited as kernels of homomorphisms from a compact Lie group G to…

A free-by-cyclic group $F_N\rtimes_\phi\mathbb{Z}$ has non-trivial centre if and only if $[\phi]$ has finite order in ${\rm{Out}}(F_N)$. We establish a profinite ridigity result for such groups: if $\Gamma_1$ is a free-by-cyclic group with…

Let $G$ be an infinite simple group of finite Morley rank and of Pr\"{u}fer $2$-rank $1$ which admits a supertight automorphism $\alpha$ such that the fixed-point subgroup $C_G(\alpha^n)$ is pseudofinite for all integers $n > 0$. We prove…

Let $G$ be a compact connected Lie group and let $P$ be a principal $G$-bundle over $K$. The gauge group of $P$ is the topological group of automorphisms of $P$. For fixed $G$ and $K$, consider all principal $G$-bundles $P$ over $K$. It is…

We prove that $\pi_1(\text{Ham}(M))$ contains an infinite cyclic subgroup, where $\text{Ham}(M)$ is the Hamiltonian group of the one point blow up of ${\Bbb C}P^3$. We give a sufficient condition for the group $\pi_1(\text{Ham}(M))$ to…

Hempel has shown that the fundamental groups of knot complements are residually finite. This implies that every nontrivial knot must have a finite-sheeted, noncyclic cover. We give an explicit bound, $\Phi (c)$, such that if $K$ is a…

Macbeath gave a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe…

Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groups which are not elementary abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results…

A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups G that admit a descending chain of proper normal finite-index subgroups, each of which is…

We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we identify conditions on automorphisms of a stable group that make it resemble…

Popov classified crystallographic complex reflection groups by determining lattices they stabilize. These analogs of affine Weyl groups have infinite order and are generated by reflections about affine hyperplanes; most arise as the…

Let $G$ be a finite group with a Sylow $p$-subgroup $P$. We prove that the principal $p$-blocks of $G$ and $N_G(P)$ are perfectly isometric under the assumption $G$ has a cyclic $p$-hyperfocal subgroup.

The structure of finite and locally finite groups in which every element has prime power order (CP-groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of…

Let $A$ be a group isomorphic with either $S_4$, the symmetric group on four symbols, or $D_8$, the dihedral group of order 8. Let $V$ be a normal four-subgroup of $A$ and $\alpha$ an involution in $A\setminus V$. Suppose that $A$ acts on a…

Let $X$ be a compact Riemann surface of genus $g\geq 2$. A cyclic subgroup of prime order $p$ of $Aut(X)$ is called properly $(p,h)$-gonal if it has a fixed point and the quotient surface has genus $h$. We show that if $p>6h+6$, then a…

Let $\Gamma=\mathrm{Cay}(G,S)$ be a Cayley digraph on a group $G$ and let $A=\mathrm{Aut}(\Gamma)$. The Cayley index of $\Gamma$ is $|A:G|$. It has previously been shown that, if $p$ is a prime, $G$ is a cyclic $p$-group and $A$ contains a…