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Gareth M. Tracey

Publications and source records attributed to Gareth M. Tracey.

6 recordsLinked to original sources
arXiv2019

Engel elements in weakly branch groups

We study properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, we prove that the set of bounded left Engel elements is always trivial in weakly branch groups. In the case of branch groups, the existence of non-trivial left Engel elements implies that these are all $p$-elements and that the group is virtually a $p$-group (and so periodic) for some prime $p$. We also show that the set of right Engel elements of a weakly branch group is trivial under a relatively mild condition. Also, we apply these results to well-known families of weakly branch groups, like the multi-GGS groups.

math.GR↗
arXiv2018

Minimal generation of transitive permutation groups

It is proved in [21] that there is a constant $c$ such that each transitive permutation group of degree $d\ge 2$ can be generated by $\lfloor cd/\sqrt{\log{d}}\rfloor$ elements. In this paper, we explicitly estimate $c$.

math.GR↗
arXiv2018

Invariable generation of permutation and linear groups

A subset $\left\{x_{1},x_{2},\hdots,x_{d}\right\}$ of a group $G$ \emph{invariably generates} $G$ if $\left\{x_{1}^{g_{1}},x_{2}^{g_{2}},\hdots,x_{d}^{g_{d}}\right\}$ generates $G$ for every $d$-tuple $(g_{1},g_{2}\hdots,g_{d})\in G^{d}$. We prove that a finite completely reducible linear group of dimension $n$ can be invariably generated by $\left\lfloor \frac{3n}{2}\right\rfloor$ elements. We also prove tighter bounds when the field in question has order $2$ or $3$. Finally, we prove that a transitive [respectively primitive] permutation group of degree $n\geq 2$ [resp. $n\geq 3$] can be invariably generated by $O\left(\frac{n}{\sqrt{\log{n}}}\right)$ [resp. $O\left(\frac{\log{n}}{\sqrt{\log{\log{n}}}}\right)$] elements.

math.GR↗