Related papers: On soluble subgroups of sporadic groups

For a variety of finite groups $\mathbf H$, let $\overline{\mathbf H}$ denote the variety of finite semigroups all of whose subgroups lie in $\mathbf H$. We give a characterization of the subsets of a finite semigroup that are pointlike…

In this paper, we study the structures of finite groups using some arithmetic conditions on the sizes of real conjugacy classes. We prove that a finite group is solvable if the prime graph on the real class sizes of the group is…

We prove a structure theorem for periodic locally soluble groups satisfying a chain condition on intersections of relatively uniformly definable subgroups using results from the theory of stable groups. The result in particular shows that…

Let $G$ be a finite insoluble group with soluble radical $ R(G)$. The solubility graph $\Gamma_{\rm S}(G)$ of $G$ is a simple graph whose vertices are the elements of $G\setminus R(G) $ and two distinct vertices $x$ and $y$ are adjacent if…

The existence of invariant transversals for a normal subgroup $H$ in a group $G$ is investigated. This yields counterexamples to a conjecture in case $H$ is abelian and $G$ is finite.

When $\mathbf h\in \mathbb Z^3$, denote by $B(X;\mathbf h)$ the number of integral solutions to the system \[ \sum_{i=1}^6(x_i^j-y_i^j)=h_j\quad (1\le j\le 3), \] with $1\le x_i,y_i\le X$ $(1\le i\le 6)$. When $h_1\ne 0$ and appropriate…

Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G. We establish criteria for N to have solvable conjugacy problem and be conjugacy separable in terms of the corresponding properties of G/N. We show that the…

It is proved that every group of the form $G=AB$ with two subgroups $A$ and $B$ each of which is either abelian or has a quasicyclic subgroup of index $2$ is soluble of derived length at most $3$. In particular, if $A$ is abelian and $B$ is…

Thompson's theorem stated that a finite group $G$ is solvable if and only if every $2$-generated subgroup of $G$ is solvable. In this paper, we prove some new criteria for both solvability and nilpotency of a finite group using certain…

We consider two decision problems in infinite groups. The first problem is Subgroup Intersection: given two finitely generated subgroups $\langle \mathcal{G} \rangle, \langle \mathcal{H} \rangle$ of a group $G$, decide whether the…

Let $1 \to N \to G \to G/N \to 1$ be a short exact sequence of countable discrete groups and let $B$ be any $G$-$C^*$-algebra. In this paper, we show that the strong Novikov conjecture with coefficients in $B$ holds for such a group $G$…

This paper helps explain the prevalence of soluble groups among the automorphism groups of regular maps (at least for `small' genus), by showing that every non-perfect hyperbolic ordinary triangle group $\Delta^+(p,q,r) = \langle\, x,y \ |…

For subsets $X,Y$ of a finite group $G$, let $Pr(X,Y)$ denote the probability that two random elements $x\in X$ and $y\in Y$ commute. Obviously, a finite group $G$ is nilpotent if and only if $Pr(P,Q)=1$ whenever $P$ and $Q$ are Sylow…

A discrete subset $S$ of a topological group $G$ is called a {\it suitable set} for $G$ if $S\cup \{e\}$ is closed in $G$ and the subgroup generated by $S$ is dense in $G$, where $e$ is the identity element of $G$. In this paper, the…

For a fixed $n\ge2$, the Houghton group $H_n$ consists of bijections of $X_n=\{1,\ldots,n\} \times \mathbb{N}$ that are `eventually translations' of each copy of $\mathbb{N}$. The Houghton groups have been shown to have solvable conjugacy…

Suppose $G$ is a finite group and $A\subseteq G$ is such that $\{gA:g\in G\}$ has VC-dimension strictly less than $k$. We find algebraically well-structured sets in $G$ which, up to a chosen $\epsilon>0$, describe the structure of $A$ and…

We show that the rational subset membership problem in $G$ can be reduced to the submonoid membership problem in $G{\times}H$ where $H$ is virtually Abelian. We use this to show that there is no algorithm reducing submonoid membership to a…

We prove that an element $g$ of prime order $>3$ belongs to the solvable radical $R(G)$ of a finite (or, more generally, a linear) group if and only if for every $x\in G$ the subgroup generated by $g, xgx^{-1}$ is solvable. This theorem…

The generalized order $e_G(g)$ of an element $g$ of a group $G$ is the smallest positive integer $k$ such that there exist $x_1,\ldots,x_k \in G$ such that $g^{x_1} \ldots g^{x_k}=1$, where $g^x=x^{-1}gx$. Let $e(G) = \max \{e_G(g)\ |\ g…

We show that if A is a finite subset of an abelian group with additive energy at least c|A|^3 then there is a subset L of A with |L|=O(c^{-1}\log |A|) such that |A \cap Span(L)| >> c^{1/3}|A|.