Related papers: Right orderable residually finite p-groups and a K…

Let p be a prime larger than 3 and let G be a finite group. We prove that G is p-solvable of p-length at most 2 if there are at most two distinct character degrees relatively prime to p in the principal p-block of G. This generalizes a…

If $p$ is an odd prime, then we prove that $\e(H_2(G,\mathbb{Z})) \mid p\ \e(G)$ for $p$ groups of class 7. We prove the same for $p$ groups of class at most $p+1$ with $\e(Z(G))=p$. We also prove Schurs conjecture if $\e(G/Z(G))$ is $2,3$…

Let $G$ be a finite group and let $\pi(G)=\{p_1, p_2, \ldots, p_k\}$ be the set of prime divisors of $|G|$ for which $p_1<p_2<\cdots<p_k$. The Gruenberg-Kegel graph of $G$, denoted ${\rm GK}(G)$, is defined as follows: its vertex set is…

In 1962, V.A. Belonogov proved that if a finite group $G$ contains two maximal subgroups of coprime orders, then either $G$ is one of known solvable groups or $G$ is simple. In this short note based on results by M. Liebeck and J. Saxl on…

We answer a question of Downey and Kurtz on left-orderable groups by showing that there is a computable left-orderable group which is not classically isomorphic to a computable group with a computable left-order.

This note proves a generalisation to inverse semigroups of Anisimov's theorem that a group has regular word problem if and only if it is finite, answering a question of Stuart Margolis. The notion of word problem used is the two-tape word…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms, we first prove some results on the solvability of finite groups in which some maximal $A$-invariant subgroups have indices a prime or the square of a…

We consider a finite group $G$ with a normal subgroup $N$ so that all elements of $G \setminus N$ have prime power order. We prove that if there is a prime $p$ so that all the elements in $G \setminus N$ have $p$-power order, then either…

We prove that the finitely generated simple left orderable groups constructed by the second author with Hyde have vanishing second bounded cohomology, both with trivial real and trivial integral coefficients. As a consequence, these are the…

We construct the first examples of an algorithmically complex finitely presented residually finite groups and first examples of finitely presented residually finite groups with arbitrarily large (recursive) Dehn function and depth function.…

In this note, we prove that for every integer $d\geq 2$ which is not a prime power, there exists a finite solvable group $G$ such that $d\mid |G|$, $\pi(G)=\pi(d)$ and $G$ has no subgroup of order $d$. We also introduce the CLT-degree of a…

It is shown that finite groups in which the order of the product of every pair of elements of co-prime order is the product of the orders, is nilpotent.

Suppose that G is a finite group and x in G has prime order p > 3. Then x is contained in the solvable radical of G if (and only if) <x,x^g> is solvable for all g in G. If G is an almost simple group and x in G has prime order p > 3 then…

We study a family of finitely generated residually finite groups. These groups are doubles $F_2*_H F_2$ of a rank-$2$ free group $F_2$ along an infinitely generated subgroup $H$. Varying $H$ yields uncountably many groups up to isomorphism.

In this note we study a class of finite groups for which the orders of subgroups satisfy a certain inequality. In particular, characterizations of the well-known groups $\mathbb{Z}_2\times\mathbb{Z}_2$ and $S_3$ are obtained.

We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by…

A group is called capable if it is a central factor group. We consider the capability of finite groups of class two and exponent $p$, $p$ an odd prime. We restate the problem of capability as a problem about linear transformations, which…

We find an example of a finite solvable group (in fact, a finite $p$-group) without any left brace structure (equiv. which is not an IYB group). Our argument is an improvement of an argument of Rump, using previous work in other areas of…

For a partition {\lambda} and a prime p, we prove a necessary and sufficient condition for there exists a composition {\delta} such that {\delta} can be obtained from {\lambda} after rearrangement and all the partial sums of {\delta} are…

We consider the capability of $p$ groups of class two and odd prime exponent. We use linear algebra and counting arguments to establish a number of new results. In particular, we settle the 4-generator case, and prove a sufficient condition…