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This is exercise 3.1.9 at page 70 of Robinson, A course in the theory of groups.

Prove that the property max-n is not inherited by normal subgroups, proceeding thus: let A be the additive group of rationals of the form $m2^{ n} , m, n \in \mathbb{ Z}$ and let $T = \langle t\rangle $ be infinite cyclic. Let t act on A by the rule $at = 2a$ . Now consider the group $ G = A \rtimes T$.

Here max-n is the following property : every ascending chain of normal subgroups becomes stationary. Of course the subgroup $A$ has not the property max-n. I can't prove that $ A \rtimes T$ has this property. Any hint ?

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Find all subgroups of $A$ that are normal in $G$ and prove that they satisfy ACC. Then consider an ascending chain of normal subgroups of $G$. The projection onto $T$ must become constant, and then the intersection with $A$ must become constant, and you are done. – Derek Holt Jan 18 '13 at 15:37
@DerekHolt what do you mean by projection onto ? – WLOG Jan 18 '13 at 21:32
he means the projection into. – YCor Jan 18 '13 at 23:34
@DerekHolt Another question: are the subgroups of A normal in G the subgroups of $\mathbb{Z}$ ? Is this correct ? – WLOG Jan 19 '13 at 9:10
No nontrivial subgroup of ${\mathbb Z}$ is normal in $G$. Let $N$ be a nontrivial normal subgroup of $G$ with $N \le A$. A nonzero element of $N$ has the form $n2^i$ with $n,i \in {\mathbb Z}$ and $n$ odd. Then by normality in $G$, we have $n2^i \in N$ for all $i \in {\mathbb Z}$. So $N$ cannot be contained in ${\mathbb Z}$, but it intersects it nontrivially. Choose the smallest odd $n \in {\mathbb Z}_{>0}$ and show that this determines $N$. – Derek Holt Jan 19 '13 at 12:30

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