Related papers: Variations on the Thompson theorem

There are several graphs defined on groups. Among them we consider graphs whose vertex set consists conjugacy classes of a group $G$ and adjacency is defined by properties of the elements of conjugacy classes. In particular, we consider…

For a finite group $G$, let $N(G)$ denote the set of conjugacy class sizes of $G$. We show that if every finite group $G$ with trivial center such that $N(G)$ equals to $N(Alt_n)$, where $n>1361$ and at least one of numbers $n$ or $n-1$ are…

Let $p$ be a prime and $\mathcal{F}$ be a saturated fusion system over a finite $p$-group $P$. The fusion system $\mathcal{F}$ is said to be nilpotent if $\mathcal{F}=\mathcal{F}_{P}(P)$. We provide new criteria for a saturated fusion…

Given a group $G$, we write $x^G$ for the conjugacy class of $G$ containing the element $x$. A famous theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the derived group…

A (discrete) group is called amenable whenever there exists a finitely additive right invariant probablity measure on it. For Thompson's group $F$ the problem whether it is amenable is a long-standing open question. We consider presentation…

In this paper, we introduce a new function related to the sum of element orders of finite groups. It is used to give some criteria for a finite group to be cyclic, abelian, nilpotent, supersolvable and solvable, respectively.

Let $G$ be a finite group and $H$ a core-free subgroup of $G$. We will show that if there exists a solvable, generating transversal of $H$ in $G$, then $G$ is a solvable group. Further, if $S$ is a generating transversal of $H$ in $G$ and…

It is proved that all finitely generated subgroups of generalized free product of two groups are finitely separable provided that free factors have this property and amalgamated subgroups are normal in corresponding factors and satisfy the…

Let $\pi$ be a set of primes containing $2$ and an odd prime $p$. It is proved that if a finite group $G$ has a Hall $\pi$-subgroup $H$, then the non-$p$-soluble length of $G$ is bounded above by the generalized Fitting height of $H$. The…

We show that the generation problem in Thompson group $F$ is decidable, i.e., there is an algorithm which decides if a finite set of elements of $F$ generates the whole $F$. The algorithm makes use of the Stallings $2$-core of subgroups of…

In this paper, we focus on a question of M. Newman on isomorphic subgroups of solvable groups. We get a reduction theorem of this question: for each prime q, assume that this question holds for every characteristic q-groups, then this…

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the…

We investigate the structure of finite groups whose non-central real class sizes have the same $2$-part. In particular, we prove that such groups are solvable and have $2$-length one. As a consequence, we show that a finite group is…

In this paper the notion of nilpotent right transversal and solvable right transversal has been defined. Further, it is proved that if a core-free subgroup has a generating solvable transversal or a generating nilpotent transversal, then…

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…

All groups have 2 generators. For every prime power q, the Generalized Burnside Theorem (Theorem GB) produces an infinite number of solvable groups, Some, such as groups of a prime power exponent, have only elements of finite order and are…

If G is a non-nilpotent group and nil(G) = {g \in G : <g, h> is nilpotent for all h\in G}, the nilpotent graph of G is the graph with set of vertices G-nil(G) in which two distinct vertices are related if they generate a nilpotent subgroup…

Following J.S. Rose, a subgroup H of a group G is said contranormal in G if G = H^G . In a certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has…

Let G be a finite group. An element x in G is a real element if x is conjugate to its inverse in G. For x in G, the conjugacy class x^G is said to be a real conjugacy class if every element of x^G is real. We show that if 4 divides no real…

In this paper, we will prove some sufficient conditions for the solvability of groups.