\emph{Addendum to} Sharply $2$-transitive groups of characteristic~$0$ [arXiv:1604.00573]

TL;DR

This paper modifies existing constructions of non-split sharply 2-transitive groups of characteristic 0 to be applicable over any field of characteristic 0, expanding the scope of prior results.

Contribution

It introduces a method to adapt the construction of sharply 2-transitive groups to arbitrary fields of characteristic 0, extending previous specific cases.

Findings

Construction applicable to any characteristic 0 field

Extension of non-split sharply 2-transitive groups

Broader applicability of existing group constructions

Abstract

In this short note we show how to modify the construction of non-split sharply $2$-transitive groups of characteristic~$0$ given by Rips and Tent [arXiv:1604.00573] to allow for arbitrary fields of characteristic 0

Full text

1. Introduction

The first sharply non-split sharply $2$-transitive groups in characteristic [math] were constructed in [RT]. However, the construction given there only works when starting from the group $AGL(1,\mathbb{Q})=\mathbb{Q}_{+}\rtimes\mathbb{Q}^{*}$. We modify the construction given there in order to prove:

Theorem 1.1**.**

For any field $\mathbb{K}$ of characteristic [math] the group $AGL(1,\mathbb{K})=\mathbb{K}_{+}\rtimes\mathbb{K}^{*}$ can be embedded into a sharply $2$-transitive group of characteristic [math] not containing any regular normal subgroup.

To prove Theorem 1.1 we introduce the following equivalence relation (replacing the equivalence relation given in [RT]): for any group $G$ and involution $j\in G$ we say that involutions $s,t\in G$ are equivalent relative to $j$ (and write $s\approx_{j}t$) if $\rm Cen(js)=\rm Cen(jt)$.

The following proposition replaces Proposition 1.3 of [RT] and provides the induction step for the proof of Theorem 1.1

Proposition 1.2**.**

Let $G$ be a group containing involutions $j,t$ and $t^{\prime}$ with $j^{t^{\prime}}=t$ and $A={\rm Cen}_{G}(j)$. Assume that $G,j,t,t^{\prime}$ and $A$ satisfy assumptions (1) – (3) of Theorem 1.1. in [RT] and furthermore:

• (4’)

for any involution $s$ with $s\approx_{j}t$ there is some $a\in A$ such that $s=t^{a}$.

• (5’)

for any involution $s\neq j$, we have ${\rm Cen}(js)=\{1\}\cup\{js^{\prime}\colon s^{\prime}\approx_{j}s\}$.

• (6’)

for any involution $s\notin AtA,s\neq j,$ there is an involution $s^{\prime}\in G$ with $s^{\prime}\approx_{j}s$ such that ${\rm Cen}(js)=\langle js^{\prime}\rangle$.

Then for any involution $v\in G$ with $v\neq j$ there exists an extension $G_{1}$ of $G$ such that for $A_{1}={\rm Cen}_{G_{1}}(j)$ there exists some $f\in A_{1}$ with $t^{f}=v$ and conditions $(1)-(3)$ and $(4^{\prime})-(6^{\prime})$ continue to hold with $G_{1}$ and $A_{1}$ in place of $G,A$.

Note that for the group $AGL(1,\mathbb{Q})$ the new equivalence relation agrees with the one given in [RT]. It is easy to see exactly as in [RT] that for any field $\mathbb{K}$ of characteristic [math], the group $AGL(1,\mathbb{K})$ satisfies properties (1) – (3) and (4’) – (6’).

Using the following lemma, the proof of Proposition 1.3 in [RT] carries over verbatim to this setting.

Lemma 1.3**.**

In the situation of Proposition 1.2 we have ${\rm Cen}(js)={\rm Cen}((js)^{n})$ for any involution $s\in G$. In particular, for involutions $s,s^{\prime}\in G$ and $n,m\in\mathbb{Z}$ such that $(js)^{n}=(js^{\prime})^{m}$ we have $s\approx_{j}s^{\prime}$.

Proof.

Since $js$ centralizes $(js)^{n}$, assumption (5’) implies $s\approx_{j}j(js)^{n}$. The second part follows directly from this. ∎

As in [RT] we also see that for any field $\mathbb{K}$ of characteristic [math], the group $AGL(1,\mathbb{K})*\mathbb{Z}$ satisfies properties (1) – (3) and (4’) – (6’). Now the proof of Theorem 1.1 follows exactly as in [RT].