7.42 (1980)

A group $U$ is called an $F_q$-group (where $q \in \pi(U)$) if, for each finite subgroup $K$ of $U$ and for any two elements $a, b$ of order $q$ in $T = N_U (K)/K$, there exists $c \in T$ such that the group $\langle a, b^c \rangle$ is finite. A group $U$ is called an $F^*$-group if each subgroup $H$ of $U$ is an $F_q$-group for every $q \in \pi(H)$ (V. P. Shunkov, 1977).
$\qquad$ a) Is every primary $F^*$-group satisfying the minimum condition for subgroups almost abelian?
$\qquad$ b) Does every $F^*$-group satisfying the minimum condition for (abelian) subgroups possess the radicable part?