Narrow Results

Let $G$ be a group and let $x\in G$ be a left $3$-Engel element of order dividing $60$. Suppose furthermore that ${〈x〉}^G$ 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.

Let G be a group and h, g ∈ G. The 2-tuple (h, g) is said to be an n-Engel pair if h = [h,_n g] and g = [g,_n h]. In this paper, we will study the subgroup generated by the n-Engel pair under certain conditions.

Let G be a group and h, g ∈ G. The 2-tuple (h, g) is said to be an n-Engel pair, n ≥ 2, if h = [h,_n g], g = [g,_n h] and h ≠ 1. In this paper, we prove that if (h, g) is an n-Engel pair, hgh^-2gh = ghg and ghg^-2hg = hgh, then n = 2k where k = 4 or k ≥ 6. Furthermore, the subgroup generated by {h, g} is determined for k = 4, 6, 7 and 8.

Let $G$ be a group and $h,g\in G$. The 2-tuple $(h,g)$ is said to be an $n$-Engel pair, $n\ge 2$, if $h=[h{,}_{n}g]$, $g=[g{,}_{n}h]$ and $h\ne 1$. Let SL$(2,F)$ be the special linear group of degree $2$ over the field $F$. In this paper, we show that given any field $L$, there is a field extension $F$ of $L$ with $[F:L]\le 6$ such that SL$(2,F)$ has an $n$-Engel pair for some integer $n\ge 4$. We will also show that SL$(2,F)$ has a $5$-Engel pair if $F$ is a field of characteristic $p\equiv \pm 1mod5$.