부산대학교 도서관

Let $x$ and $y$ be elements of a group $G$ and let $n$ be a positiveinteger. The commutator $[x,_ny]$ is defined by the rules $[x,_0y]=x$,$[x,_1y]=[x,y]=x^{-1}y^{-1}xy$ and ${[x,_{n+1}y] = [[x,_ny],y]}$ for all$n>0$. The element $x$ is a left $n$-Engel element if for all $y$ in$G$ the commutator $[y,_nx]=1$.\par Suppose $\germ{w} = \germ{w}(x_1,\dots, x_k)$ is a group word ofweight $k$. The $\germ{w}$-values of $G$ is the set $G_\germ{w}$consisting of all the values we obtain by evaluating $\germ{w}$ forevery $k$-tuple of elements of $G$. The subgroup $\germ{w}(G)$ is theverbal subgroup of $G$ generated by $G_\germ{w}$.\par Let $x_1,x_2,\dots $ be variables. The weight of a group word$\germ{w}$ is the number of variables in $\germ{w}$. The uniquemultilinear or outer commutator word of weight one is $\gamma_1$ where${\gamma_1(x_1)=x_1}$. The word $\theta$ is a multilinear commutator wordof weight $n>1$, if ${\theta=[\theta_1,\theta_2]}$, where $\theta_1$ and$\theta_2$ are multilinear words of weight $n_1$ and $n_2$respectively, and ${n=n_1+n_2}$. By definition, each variable of amultilinear commutator word appears at most once.\par The paper under review is a survey of results published since theEngel groups theme day which was part of Groups St. Andrews 2009 inBath. These results focus on verbal generalizations of Engel groups. Welist two representative theorems presented in the paper and refer thereader back to the paper for many more results and further details.\par Theorem. Let $m$, $n$ be positive integers, $\germ{v}$ a multilinearcommutator word and ${\germ{w} = \germ{v}^m}$.\roster\item"(i)" If $G$ is a locally graded group in which all $\germ{w}$-values areleft $n$-Engel elements, then the verbal subgroup $\germ{w}(G)$ islocally nilpotent.\item"(ii)" The class of all groups $G$ in which the $\germ{w}$-values are left$n$-Engel elements and the verbal subgroup $\germ{w}(G)$ is locallynilpotent is a variety.\endroster\par The word $\germ{w}$ is called semiconcise if the finiteness of$G_\germ{w}$ for a group $G$ always implies the finiteness of thesubgroup $[\germ{w}(G), G]$.\par Theorem. For any positive integer $n$, the $n$-Engel word $[x,_n y]$is semiconcise.