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2.22 (1966)

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An abstract group-theoretic property $\Sigma$ is said to be radical (in our sense) if, in any group $G$, the subgroup $\Sigma(G)$ generated by all normal $\Sigma$-subgroups is a $\Sigma$-subgroup itself (called the $\Sigma$-radical of $G$). A radical property $\Sigma$ is said to be strongly radical if, for any group $G$, the factor-group $G/\Sigma(G)$ contains no non-trivial normal $\Sigma$-subgroups.
$\qquad$ a) Is the property $\overline{RN}$ radical? strongly radical?
$\qquad$ b) Is the property $N^0$ (see 1.38) radical?

Contributor: Sh. S. Kemkhadze

Progress

b) No. The group $\text{SL}_n(\mathbb{Z})$ for sufficiently large $n \geqslant 3$ contains a non-abelian finite simple group and therefore is not an $N^0$-group. On the other hand, as shown in (M. I. Kargapolov, Yu. I. Merzlyakov, in: Itogi Nauki. Algebra. Topologiya. Geometriya, 1966, VINITI, Moscow, 1968, 57–90 (Russian)), it is the product of its congruence-subgroups mod 2 and mod 3, which have central systems and hence are $N^0$-groups (see problem 1.38). (Yu. I. Merzlyakov, 1973.)

a) remains open

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