Some simplifications in the proof of the Sims conjecture
We prove an elementary lemma concerning primitive amalgams and use it to greatly simplify the proof of the Sims conjecture in the case of almost simple groups.
math.GR↗
arXiv subjects
Gareth Tracey
Publications and source records attributed to Gareth Tracey.
On the generalized Fitting height and insoluble length of finite groups
We prove two conjectures of E. Khukhro and P. Shumyatsky concerning the Fitting height and insoluble length of finite groups. As a by-product of our methods, we also prove a generalization of a result of Flavell, which itself generalizes Wielandt's Zipper Lemma and provides a characterization of subgroups contained in a unique maximal subgroup. We also derive a number of consequences of our theorems, including some applications to the set of odd order elements of a finite group inverted by an involutory automorphism.
math.GR↗
Left $3$-Engel elements in groups of exponent $60$
Let $G$ be a group and let $x\in G$ be a left $3$-Engel element of order dividing $60$. Suppose furthermore that $\langle x\rangle^{G}$ has no elements of order $8$, $9$ and $25$. We show that $x$ is then contained in the locally nilpotent radical of $G$. In particular all the left $3$-Engel elements of a group of exponent $60$ are contained in the locally nilpotent radical.
math.GR↗
An upper bound on the Chebotarev invariant of a finite group
A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. The first author recently showed that $C(G)\le β\sqrt{|G|}$ for some absolute constant $β$. In this paper we show that, when $G$ is soluble, then $β$ is at most $5/3$. We also show that this is best possible. Furthermore, we show that, in general, for each $ε>0$ there exists a constant $c_ε$ such that $C(G)\le (1+ε)\sqrt{|G|}+c_ε$.
math.GR↗
Finite groups with large Chebotarev invariant
A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ is said to invariably generate $G$ if the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. The authors recently showed that for each $ε>0$, there exists a constant $c_ε$ such that $C(G)\le (1+ε)\sqrt{|G|}+c_ε$. This bound is asymptotically best possible. In this paper we prove a partial converse: namely, for each $α>0$ there exists an absolute constant $δ_α$ such that if $G$ is a finite group and $C(G)>α\sqrt{|G|}$, then $G$ has a section $X/Y$ such that $|X/Y|\geq δ_α\sqrt{|G|}$, and $X/Y\cong \mathbb{F}_q\rtimes H$ for some prime power $q$, with $H\le \mathbb{F}_q^{\times}$.
math.GR↗
Generating maximal subgroups of finite almost simple groups
For a finite group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. In this paper, given a finite almost simple group $G$ and any maximal subgroup $H$ of $G$, we determine a precise upper bound for $d(H)$. In particular, we show that $d(H)\leq 5$, and that $d(H)\geq 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.
math.GR↗