The biopsychosocial model remains a key paradigm for healthcare, despite widely recognised scientific and philosophical shortcomings. Here we report on recent updates integrating evolutionary theory with the biopsychosocial model to provide a more comprehensive and scientifically complete approach to understanding the multiple relevant levels of causation of medical and psychiatric problems.

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.

We show that if $G$ is a group of exponent 5, and if $G$ satisfies the Engel-4 identity $[x,y,y,y,y]=1$, then $G$ is locally finite. By a result of Traustason, this implies that Engel-4 5-groups are locally finite. We also show that a group of exponent 5 is locally finite if and only if it satsifies the identity

$$[x,[y,z,z,z,z],[y,z,z,z,z]]=1.$$

This result implies that a group of exponent 5 is locally finite if its three generator subgroups are finite.

1991 Mathematics Subject Classification: 20D15, 20F45, 20F50.