부산대학교 도서관

Let $x$ and $y$ be elements of a group $G$ and let $n$ be a positive integer. 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 of weight $k$. The $\germ{w}$-values of $G$ is the set $G_\germ{w}$ consisting of all the values we obtain by evaluating $\germ{w}$ for every $k$-tuple of elements of $G$. The subgroup $\germ{w}(G)$ is the verbal 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 unique multilinear or outer commutator word of weight one is $\gamma_1$ where ${\gamma_1(x_1)=x_1}$. The word $\theta$ is a multilinear commutator word of 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 a multilinear commutator word appears at most once. \par The paper under review is a survey of results published since the Engel groups theme day which was part of Groups St. Andrews 2009 in Bath. These results focus on verbal generalizations of Engel groups. We list two representative theorems presented in the paper and refer the reader back to the paper for many more results and further details. \par Theorem. Let $m$, $n$ be positive integers, $\germ{v}$ a multilinear commutator word and ${\germ{w} = \germ{v}^m}$. \roster \item"(i)" If $G$ is a locally graded group in which all $\germ{w}$-values are left $n$-Engel elements, then the verbal subgroup $\germ{w}(G)$ is locally 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 locally nilpotent 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 the subgroup $[\germ{w}(G), G]$. \par Theorem. For any positive integer $n$, the $n$-Engel word $[x,_n y]$ is semiconcise.