A right Engel sink of an element g of a group G is a set ${\mathscr R}(g)$ such that for every x ∈ G all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, g is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) It is proved that if every element of a compact (Hausdorff) group G has a countable right Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.

Let $g$ be an element of a finite group $G$ and let $R_{n}(g)$ be the subgroup generated by all the right Engel values $[g,_{n}x]$ over $x\in G$ . In the case when $G$ is soluble we prove that if, for some $n$ , the Fitting height of $R_{n}(g)$ is equal to $k$ , then $g$ belongs to the $(k+1)$ th Fitting subgroup $F_{k+1}(G)$ . For nonsoluble $G$ , it is proved that if, for some $n$ , the generalized Fitting height of $R_{n}(g)$ is equal to $k$ , then $g$ belongs to the generalized Fitting subgroup $F_{f(k,m)}^{\ast }(G)$ with $f(k,m)$ depending only on $k$ and $m$ , where $|g|$ is the product of $m$ primes counting multiplicities. It is also proved that if, for some $n$ , the nonsoluble length of $R_{n}(g)$ is equal to $k$ , then $g$ belongs to a normal subgroup whose nonsoluble length is bounded in terms of $k$ and $m$ . Earlier, similar generalizations of Baer’s theorem (which states that an Engel element of a finite group belongs to the Fitting subgroup) were obtained by the first two authors in terms of left Engel-type subgroups.

For an element g of a group G, an Engel sink is a subset ${\mathscr E}$(g) such that for every x ∈ G all sufficiently long commutators [. . .[[x, g], g], . . ., g] belong to ${\mathscr E}$(g). A finite group is nilpotent if and only if every element has a trivial Engel sink. We prove that if in a finite group G every element has an Engel sink generating a subgroup of rank r, then G has a normal subgroup N of rank bounded in terms of r such that G/N is nilpotent.

Suppose that a finite group G admits an automorphism of order 2 n such that the fixed-point subgroup of the involution is nilpotent of class c. Let m = ) be the number of fixed points of . It is proved that G has a characteristic soluble subgroup of derived length bounded in terms of n, c whose index is bounded in terms of m, n, c. A similar result is also proved for Lie rings.

Quantifying resilience allows for several testable hypotheses, such as that resilience is equal to the number of mental health problems given a known quantity of stressor load. The proposed model lends itself well to prospective studies with data collection pre- and post-adversity; however, prestressor assessments are not always available. Challenges remain for adapting quantifying resilience to animal research, even if the idea of its translation value is significant.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the verbal subgroup $w(G)$ is locally finite.

We show that if a periodic residually-finite group G has a 2-element with finite centralizer then G is locally finite.