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I am looking for an infinite normal subgroup s.t. for $N \unlhd G$ we have $N \cap Soc(G) \not\subset G$.

It must be infinite, since we proved in the lecture, that for finite $N$, $N \cap Soc(G)$ is always contained in $Soc(N)$.

I have now found $S_{3}\times \mathbb{Z}$ then $S_{3} \times \{0\}$ is a normal subgroup.

How can I calculate the Socle of this group and normal subgroup?

Best, Kathrin!

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Do you mean $N \cap {\rm Soc}(G) \not\subset {\rm Soc}(N)$? You say you are looking for an infinite normal subgroup, but then you give an example of a finite normal subgroup, so it is not clear what you are trying to do. The socle of $S_3$ is $A_3$ of order 3. – Derek Holt Nov 22 '12 at 10:23
What is the proof that $N\cap \mbox{Soc}(G)\subseteq \mbox{Soc}(N)$ for finite groups? At which step would it fall apart if we didn't require the groups were finite? – Alexander Gruber Nov 22 '12 at 20:39

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