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I would like to find an exact sequence $1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1$ of groups where $N,Q$ are supersolvable and G is not. I found it but in my example $G$ is finite, I was wondering if you have a nice example with $G$ an infinite group, I tried with a lot of groups without finding it. Do you have any idea?

EDIT: Suppose that $G$ is a finite not supersolvable group and we have an exact sequence $1\rightarrow K\rightarrow G\rightarrow G/K\rightarrow 1$ with $K$ and $G/K$ supersolvable. Define $G^\prime=G\times\mathbb{Z}$, then $G^\prime$ is not supersolvable because if I have a normal-cyclic series the I project it on $G$ and I obtain a normal-cyclic series for $G$ and this is a contradiction. So I can take $1\rightarrow K\times\{0\}\rightarrow G\rightarrow G/(K\times\{0\})\rightarrow1$. Now $K\times\{0\}$ is supersolvable and $G/(K\times\{0\})\cong (G/K)\times\mathbb{Z}$ and it's supersolvable, am I right?

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Couldn't you enlarge your class of examples by taking a direct sum with any finitely generated abelian group? –  jspecter Feb 24 '12 at 3:06

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up vote 3 down vote accepted

If you want a more essentially infinite example, you could take a semidirect product of ${\mathbb Z}^2$ by ${\mathbb Z}$ with an irreducible action. For example $\langle x,y,z \mid xy=yx, z^{-1}xz=y, z^{-1}yz=xy \rangle$.

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how did you create this example? –  John Feb 28 '12 at 2:48

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