I am looking for examples of socle and normal-subgroup relations.
If $G=S_{4}$, the normal subgroups are $A_{4}$ and $V_{4}$, thus the socle of $G$ is the klein-four group, and for $A_{4} \cap Soc(G) = Soc(A_{4}) = V_{4}$.
Now I am looking for some easy example where this does not work. Where we have no normal subgroup in $G$, s.t. the cut with the $Soc(G)$ equals the socle of this normal subgroup.
Any hints where I can look for such a group?
Thanks and best :) Kathrin
$G = \{e, T, T^{2}, T^{3}, S, ST, ST^{2}, ST^{3}\}$ where $T$ is a turn of $\frac{\pi}{2}$ and $S$ the mirrow.
Then I have the normal subgroups: $e,
,$. The minimal normal subgroup is right? as this is contained in all the other normal subgroups. So $Soc(G) = = \{e, T^{2}\}$. But now for all normal subgroups the intersection is $$. But isn't also the Socle of all these normal subgroups $$? $ for example, the minimal normal subgroups are $$ and $ – 2012-11-21$ s.t $Soc(N) = \{e, S, T^{2}, ST^{2}\}$? Thanks a lot for all the help :)