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John S. Wilson

Publications and source records attributed to John S. Wilson.

10 recordsLinked to original sources
arXiv2026

Uncountably many local isomorphism types of compactly generated simple groups

A major open question in the theory of locally compact groups is the following. Let $\mathscr{S}$ be the class of non-discrete compactly generated totally disconnected locally compact groups that are topologically simple. Is the number of local isomorphism classes of groups in $\mathscr{S}$ uncountable? We have answered this question, showing that there are $2^{\aleph_0}$ local isomorphism classes in $\mathscr{S}$. This preprint is an overview of our forthcoming paper. Our result was obtained without the use of artificial intelligence; it arose from a problem session that ran over several days at the workshop "Branch groups: subgroups, rigidity, topologies" at the Universidad Complutense de Madrid, organised by Dominik Francoeur, Alejandra Garrido and Tatiana Nagnibeda.

math.GR↗
arXiv2022

Profinite groups with few conjugacy classes of $p$-elements

It is proved that a profinite group $G$ has fewer than $2^{\aleph_0}$ conjugacy classes of $p$-elements for an odd prime $p$ if and only if its $p$-Sylow subgroups are finite. (Here, by a $p$-element one understands an element that either has $p$-power order or topologically generates a group isomorphic to ${\mathbb Z}_p$.) A weaker result is proved for $p=2$.

math.GR↗
arXiv2021

The soluble radical and orbits of certain maps on finite groups

For each element $u$ in a finite group $G$ define a map $θ_u\colon G\to G$ by $θ_u(g)=[g^{-u},g]$ and set $Θ_G(u)=\{g\in G\mid θ_u^n(g)=g \hbox{ for some } n>0\}$. Then $θ_u$ induces a permutation of $Θ_G(u)$; let $β_G(u)$ be the number of orbits apart from $\{1\}$. Building on work of J.N. Bray, R.A. Wilson and the second author, we show that the index of the soluble radical of a finite group $G$ is bounded in terms of the values of $β_G(u)$ for $2$-elements $u$.

math.GR↗
arXiv2020

First-order recognisability in finite and pseudofinite groups

It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with 'solubility' replaced by 'nilpotence' and 'perfectness', among others, are false. These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini's theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.

math.GR↗
arXiv2017

Recognizing the real line

Let $(Ω, \leq)$ be a totally ordered set. We prove that if Aut$(Ω,\leq)$ is transitive and satisfies the same first-order sentences as the automorphism group of the real line (in the language of groups) then $Ω$ and and the real line are isomorphic ordered sets. This improvement of a theorem of Gurevich and Holland is obtained as a consequence of a study of centralizers associated with certain transitive subgroups of Aut$(Ω,\leq)$.

math.GR↗
arXiv2016

The first-order theory of $\ell$-permutation groups

Let $(Ω, \leq)$ be a totally ordered set. We prove that if $\Aut(Ω,\leq)$ is transitive and satisfies the same first-order sentences as $\Aut(\RR,\leq)$ (in the language of lattice-ordered groups) then $Ω$ and $\RR$ are isomorphic ordered sets. This improvement of a theorem of Gurevich and Holland is obtained as one of many consequences of a study of centralizers and coloured chains associated with certain transitive subgroups of $\Aut(Ω,\leq)$.

math.GR↗
arXiv2014

Residual nilpotence and ordering in one-relator groups and knot groups

Let $G=< x,t\mid w>$ be a one-relator group, where $w$ is a word in $x,t$. If $w$ is a product of conjugates of $x$ then, associated with $w$, there is a polynomial $A_w(X)$ over the integers, which in the case when $G$ is a knot group, is the Alexander polynomial of the knot. We prove, subject to certain restrictions on $w$, that if all roots of $A_w(X)$ are real and positive then $G$ is bi-orderable, and that if $G$ is bi-orderable then at least one root is real and positive. This sheds light on the bi-orderability of certain knot groups and on a question of Clay and Rolfsen. One of the results relies on an extension of work of G. Baumslag on adjunction of roots to groups, and this may have independent interest.

math.GR↗
arXiv2014

Metric ultraproducts of finite simple groups

Some new results on metric ultraproducts of finite simple groups are presented. Suppose that G is such a group, defined in terms of a non-principal ultrafilter ω on N and a sequence {(G_i)_{i \in N}} of finite simple groups, and that G is neither finite nor a Chevalley group over an infinite field. Then G is isomorphic to an ultraproduct of alternating groups or to an ultraproduct of finite simple classical groups. The isomorphism type of G determines which of these two cases arises, and, in the latter case, the ω-limit of the characteristics of the groups Gi. Moreover G is a complete path-connected group with respect to the natural metric on G.

math.GR↗
arXiv2013

On subgroups of finite index in branch groups

We give a structural description of the normal subgroups of subgroups of finite index in branch groups in terms of rigid stabilizers. This gives further insight into the structure lattices of branch groups introduced by the second author. We derive a condition concerning abstract commensurability of branch groups acting on the p-ary tree for any prime p.

math.GR↗