6.49 (1978)

No, not always. Let $G = [(A \times B \times C \times D) \rtimes (\langle g \rangle \times \langle t \rangle)] \rtimes (Y \times \langle x \rangle)$, where $A, B, C, D, Y$ are quasicyclic $p$-groups, $x^2 = 1$, while $g$ and $t$ are of infinite order. Let $A = \bigcup_{n=1}^\infty \langle a_n \rangle$, $B = \bigcup_{n=1}^\infty \langle b_n \rangle$, $C = \bigcup_{n=1}^\infty \langle c_n \rangle$, $D = \bigcup_{n=1}^\infty \langle d_n \rangle$, $Y = \bigcup_{n=1}^\infty \langle y_n \rangle$ with $a_{n+1}^p = a_n$, $b_{n+1}^p = bn, c_{n+1}^p = cn, d_{n+1}^p = dn, y_{n+1}^p = y_n$. We impose the relations $[ABCD, Y] = [ABC, g] = [ABD, t] = 1$; $[x, g] = gt^{-1}$; $[x, a_n] = a_n b_n^{-1}$; $[x, c_n] = c_n d_n^{-1}$; $[g, d_n] = b_n$; $[t, c_n] = a_n$; $[y_n, g] = c_n$; $[y_n, t] = d_n$. Then all abelian normal subgroups of $G$ satisfy the minimal condition for subgroups. The subgroup $H = [(A \times B \times C \times D) \rtimes (\langle g \rangle \times \langle t \rangle)] \rtimes Y$ does not satisfy the minimal condition for abelian normal subgroups, since the subgroups $E_n = A \times B \times C \times \langle g^{2^n} \rangle$ are normal in $H$ and form a strictly decreasing chain. (S. A. Chechin, Abstracts of 15th All-USSR Algebraic Conf., Krasnoyarsk, 1979).