Consider $V_*$, the subgroup of R. Thompson's group $V$ which stabilises $0^\omega$ under the natural action upon the Cantor set, $\{0, 1\}^\omega$. Let $G_*$ be any subgroup of a finitely generated group $G$. We show that $G_*$ is a context-free subgroup of $G$ if and only if $G_*$ is a pullback of $V_*$ under a homomorphism $G \rightarrow V$. In particular, this shows the existence of a hardest context-free membership problem. As a consequence, we prove that a group $G$ embeds into $V$ if and only if it is the transition group of a finite union of context-free automata, or equivalently, if there exist finitely many context-free subgroups of $G$ whose cores intersect trivially.
For a subset $S$ of a finite group $G$, let $I_G(S)$ denote the set of commutators $[g,x]=g^{-1}g^x$, where $g\in G$ and $x\in S$. Suppose that a finite group $G$ has a Carter subgroup $C$, that is, a nilpotent subgroup containing its normalizer. Suppose that any subgroup generated by a subset of $I_G(C)$ is $r$-genera...
Cristina Acciarri, R. Guralnick, Evgeny Khukhro et al.· 0 citations
A subgroup $H$ of Thompson's group $F$ is closed if every piecewise-$H$ function in $F$ belongs to $H$. Prominent examples of closed subgroups are the maximal subgroups of infinite index, stabilizers and pointwise stabilizers of sets of points, and many of Jones'subgroups. A closed subgroup is finitely generated if and...
A subset $X$ of a finite group $G$ is $k$-power-avoiding if for all $g\in G$ we have that $\{g,g^k\}\not\subseteq X$. The paper shows that if $G$ contains a $k$-power-avoiding subset $X$ with $|X|\geq |G|-c$, then the group $G^k$ generated by the $k$th powers of elements of $G$ has bounded order (in $k$ and $c$). We pr...
S. Blackburn, S. Hart, Daniel McVeagh· 0 citations
For a finite group $H$, let $\nu(H)$ denote the maximum order of a nilpotent subgroup of $H$. We prove that every finite solvable transitive permutation group $P$ on a finite set $\Omega$ has a subset $\Delta\subseteq\Omega$ such that $|P:P_\Delta|\ge\nu(P_\Delta)$. We also prove that if a finite solvable group $H$ act...
Let $k$ be a field, let $V=k^n$, and let $G$ be a classical group acting on $V$. For a finite subset $E\subset V$ and a basis $u=(u_1,\dots,u_n)$ of $V$, we study the set of $G$-frames of type $u$ contained in $E$, or, equivalently, the set $T_{E,u}^G:=\{g\in G(k)\ |\ g u_i\in E\text{ for each $i$}\}$. In the cases des...
We prove a graph-theoretical characterisation of finitely generated subgroups of Thompson's group $V$: a finitely generated group embeds in $V$ if and only if it admits a faithful context-free action, or equivalently if it belongs to the class CF-TR of transition groups of context-free graphs recently introduced by Mat...
Corentin Bodart, D. D’Angeli, Davide Perego et al.· 1 citation
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