  Let $w = w(x_1,..., x_n)$ be a word, i.e. an element of the free group $F
=<x_1,...,x_n>$ on $n$ generators $x_1,..., x_n$. The verbal subgroup $w(G)$ of
a group $G$ is the subgroup generated by the set $\{w (g_1,...,g_n)^{\pm 1} |
g_i \in G, 1\leq i\leq n \}$ of all $w$-values in $G$. We say that a (finite)
group $G$ is $w$-maximal if $|G:w(G)|> |H:w(H)|$ for all proper subgroups $H$
of $G$ and that $G$ is hereditarily $w$-maximal if every subgroup of $G$ is
$w$-maximal. In this text we study $w$-maximal and hereditarily $w$-maximal
(finite) groups.
