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I've got the following exercise to solve:
Let $G$ be a polycyclic group, $N \lhd G$ and let $h(\cdot)$ be the Hirsch length. Then $h(G) = h(N) + h(G/N)$
I know that subgroups and quotients of polycyclic groups are polycyclic and I know how the corresponding polycyclic series look like. This is probably not too hard, but right now I honestly have no clue how to tackle this problem. I'd be happy if anyone could point me in the right direction.

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Since the Hirsch length does not depend on the choice of subnormal series, can't you choose your subnormal series $$G=G_0\supset G_1\supset\cdots\supset G_n=0$$ such that $G_m=N$ for some $m$? Then $G_m\supset\cdots\supset 0$ is a subnormal series for $N$ and $G_0/N\supset\cdots\supset G_m/N=0$ is a subnormal series for $G/N$. Clearly their lengths sum to $n$. – Alex Becker Jan 15 '12 at 20:09

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