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!