13.
A group
$G$ is said to be a minimax group if it has a finite series whose factors satisfy either the minimal or the maximal condition. Let
$D(G)$ denotes the subgroup of
$G$ generated by all the Chernikov divisible normal subgroups of
$G$ . If
$G$ is a soluble-by-finite minimax group and if
$D(G)=1$ , then
$G$ is said to be a reduced minimax group. Also
$G$ is said to be an
$ M_{r}C$ -group (respectively,
$PC$ -group), if
$G/C_{G} \left(x^{G}\right)$ is a reduced minimax (respectively, polycyclic-by-finite) group for all
$x\in G$ . These are generalisations of the familiar property of being an
$FC$ -group. Finally, if
$\mathfrak X $ is a class of groups, then
$G$ is said to be a minimal non-
$\mathfrak X $ -group if it is not an
$\mathfrak X $ -group but all of whose proper subgroups are
$\mathfrak X $ -groups. Belyaev and Sesekin characterized minimal non-
$FC$ -groups when they have a non-trivial finite or abelian factor group. Here we prove that if
$G$ is a group that has a proper subgroup of finite index, then
$G$ is a minimal non-
$M_{r}C$ -group (respectively, non-
$PC$ -group) if, and only if,
$G$ is a minimal non-
$FC$ -group.
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